Uiua, 131 bytes

⊃(⊢⊚=@R|¬∊" K"|⊢⊚=@K)⊜∘⊸≠@\n
◇⧈-⊙◌⊢path(≡/+⌵⟜+⌟⊂⊂⊂∩¯⊙:⊙⊙∩≡⇌∩₄(≡⊂0⍜¯⇡⊢⊚)∩⊃(⇌↙|↘+₁)⊙::⟜₂(⊓⌟⊏≡⊏)⤚°⊟)𝄐≍
≡&p⍚$"_ _"⊃(⊏⊙"ESWN"⊗±⊙A₂|/+⍉⌵)

Newlines are unnecessary, and are present for readability

Test suite

Python3, 472 bytes

E=enumerate
S=lambda x:(len(x[2]),sum(a for _,a in x[2]))
def f(b):
 d,D={(x,y):v for x,r in E(b)for y,v in E(r)},{}
 x,y=[i for i in d if'R'==d[i]][0]
 q=[(x,y,[],[])]
 while q:
  (x,y,p,P),*q=q
  if'K'==d[(x,y)]:return p
  for X,Y in[(0,1),(1,0),(-1,0),(0,-1)]:
   U=(*(C:=(x+X,y+Y)),p+[((X,Y),1)]if[]==p or p[-1][0]!=(X,Y)else p[:-1]+[((X,Y),p[-1][-1]+1)],P+[C])
   if d.get(C)in[' ','K']and C not in P and(C not in D or D[C]>=S(U)):q+=[U];D[C]=S(U)
  q=sorted(q,key=S)

Try it online!

Python 2.6 (535 characters)

exec 'eJzNlLFugzAQhneewhki2/KBworksVOkDu3QgVqVE2hBioFgCDhV371nJ1VbKR3SKYtl/jvs77MwtenafiBVqbs9WGcjLW05MB69tj05QEfqhpTNaMpeDyXDhsQORd3wLCK+YwT7u6PzFVK/Ep9Tsn6gmU50UTA2woHzc01KuqYZ20PPZSh85/iCO2etzBnTG8tcvlLxnovTPFVxzyGEC4lpiBay5xiuYMXBcRVtzxqTfP+IpqrelaRFsheoYJbBNvFj13asxd23gXHGmZU7bTaFDgiZH+MUYydtKBuZRuS0nvPmOt564Sl3CmlxcWAG6D3lXIkpeUMGB7nyfj82HW3FWvjTTVSYCXNJEUupEimannu+nl04WyM8XoB1F13E9S6Pt+ki0vDZXOdyd5su8X9cnm7DBa/tLGW4yh7yKCn1rIF+9vSTj/HiBeqCS1M3bMrnwOvbl5Ysi+eGLlkBhosjxl1fNwM5Ak7xH6CiT3SdT4U='.decode('base64').decode('zlib')

Unpacks to a severely mistreated A* implementation. Reads stdin. Searches for solutions with minimal total distance. Breaks ties by preferring minimal number of directions. Lists moves to stdout. Finds kittens.

Unpacked:

The source has been manually anti-golfed in a few places in order to obtain a smaller compressed representation. For example, a for loop over the compass directions was unrolled.

import heapq,sys 
a=set() 
for v,p in enumerate(sys.stdin): 
 for u,s in enumerate(p): 
  if s in' KR':a.add((u,v)) 
  if s=='K':(q,r)=(u,v) 
  if s=='R':y=(u,v) 
o=[((abs(y[0]-q)+abs(y[1]-r),(y[0]!=q)+(y[1]!=r)),(0,0),y)] 
c=set() 
w={} 
while o: 
 _,h,x=heapq.heappop(o) 
 c.add(x) 
 s=lambda(u,v):(u,v-1) 
 y=s(x) 
 m=1 
 while y in a-c: 
  w[y]=[(h,x,(m,'N'))]+w.get(y,[]) 
  heapq.heappush(o,((abs(y[0]-q)+abs(y[1]-r)+h[0]+m,(y[0]!=q)+(y[1]!=r)+h[1]+1),(h[0]+m,h[1]+1),y)) 
  m+=1 
  y=s(y) 
 s=lambda(u,v):(u,v+1) 
 y=s(x) 
 m=1 
 while y in a-c: 
  w[y]=[(h,x,(m,'S'))]+w.get(y,[]) 
  heapq.heappush(o,((abs(y[0]-q)+abs(y[1]-r)+h[0]+m,(y[0]!=q)+(y[1]!=r)+h[1]+1),(h[0]+m,h[1]+1),y)) 
  m+=1 
  y=s(y) 
 s=lambda(u,v):(u+1,v) 
 y=s(x) 
 m=1 
 while y in a-c: 
  w[y]=[(h,x,(m,'E'))]+w.get(y,[]) 
  heapq.heappush(o,((abs(y[0]-q)+abs(y[1]-r)+h[0]+m,(y[0]!=q)+(y[1]!=r)+h[1]+1),(h[0]+m,h[1]+1),y)) 
  m+=1 
  y=s(y) 
 s=lambda(u,v):(u-1,v) 
 y=s(x) 
 m=1 
 while y in a-c: 
  w[y]=[(h,x,(m,'W'))]+w.get(y,[]) 
  heapq.heappush(o,((abs(y[0]-q)+abs(y[1]-r)+h[0]+m,(y[0]!=q)+(y[1]!=r)+h[1]+1),(h[0]+m,h[1]+1),y)) 
  m+=1 
  y=s(y) 
 if x==(q,r): 
  z='' 
  while x in w: 
   _,x,(m,d)=min(w[x]) 
   z='%s %d\n'%(d,m)+z 
  print z, 
  o=[]

Python 2.6 (504 characters)

import sys,itertools
W,Q=len,list   
M=[]
B=[]
for l in sys.stdin.readlines():
    r=Q(l.strip())
    B.append([0]*W(r))
    M.append(r)
    if "R" in r:x,y=r.index("R"),W(B)-1
def S(B,M,x,y,P):
    c=M[y][x]
    b=B[y][x]
    if b and W(P)>b:return 0
    B[y][x]=W(P)
    if c=="K":return P  
    elif c=="R" and P:return 0
    if c in "R ":
        b=[]
        for q,w,s in((1,0,"E"),(-1,0,"W"),(0,-1,"N"),(0,1,"S")):
            r=S(B,M,x+q,y+w,P+s)
            if r and(W(r)<W(b)or not b):b=r
        if b:return b
    return 0
print "\n".join(k+str(W(Q(g)))for k,g in itertools.groupby(S(B,M,x,y,"")))

Scala 2.8 (451 characters)

...but it doesn't solve ties in favor of least amount of directions (though it does find the least amount of steps).

val m=io.Source.stdin.getLines.map(_.toArray).toSeq
var l=m.indexWhere(_ contains'R')
var c=m(l)indexOf'R'
val q=collection.mutable.Queue(l->c->"")
def s{val((l,c),p)=q.dequeue
if("R "contains m(l)(c))for((i,j,k)<-Seq((-1,0,"N"),(0,1,"E"),(1,0,"S"),(0,-1,"W")))q.enqueue(((l+i,c+j),p+k))
m(l)(c)='X'}
def P(s:String){if(s.nonEmpty){val (h,t)=s.span(s.head==)
println(s.head+" "+h.size)
P(t)}}
def Q{s
val((l,c),p)=q.head
if (m(l)(c)=='K')P(p)else Q}
Q

Scala 2.8, 642 characters, solves ties correctly;

val m=io.Source.stdin.getLines.toSeq
var b=Map(0->0->0).withDefault(_=>Int.MaxValue)
var l=m.indexWhere(_ contains'R')
var c=m(l)indexOf'R'
val q=collection.mutable.PriorityQueue(l->c->List((' ',0)))(Ordering.by(t=>(-t._2.map(_._2).sum,-t._2.size)))
def s{val(p@(l,c),u@(h@(d,n))::t)=q.dequeue
if("R ".contains(m(l)(c))&&u.map(_._2).sum<=b(p)){b=b.updated(p,u.map(_._2).sum)
for((i,j,k)<-Seq((-1,0,'N'),(0,1,'E'),(1,0,'S'),(0,-1,'W'))){
val o=if(k==d)(d,n+1)::t else (k,1)::h::t
q.enqueue(((l+i,c+j),o))}}}
def P(l:List[(Char,Int)]){l.reverse.tail.foreach{t=>println(t._1+" "+t._2)}}
def Q{s;val((l,c),p)=q.head;if (m(l)(c)=='K')P(p)else Q}
Q

It discovered a shorter path for the second example than the one given in the problem:

N 1
E 10
N 4
E 2

Ruby - 539 characters

Could do with a lot of improvement, but it does work for shortest steps as well as directions.

M=[*$<]
r=M.map{|q|q.index('R')||0}
k=M.map{|q|q.index('K')||0}
D=M.map{|q|q.split('').map{[99,[]]}} 
def c h 
h.map{|i|i.inject([[]]){|a,b|a.last[0]!=b ? a<<[b, 1]:a.last[1]+=1;a}}.sort_by{|a|a.length}[0]
end
def t x,y,s,i
z,w=D[x][y][0],D[x][y][1]
if [' ','R','K'].index(M[x][y, 1])&&(z>s||z==s&&c(w).length>=c([i]).length)
D[x][y]=[s,z==s ? w<<i:[i]]
s+=1
t x+1,y,s,i+['S']
t x-1,y,s,i+['N']
t x,y+1,s,i+['E']
t x,y-1,s,i+['W']
end
end
t r.index(r.max), r.max, 0, []
puts c(D[k.index(k.max)][k.max][1]).map{|a|a*''}

C++ 1002 899 799chars

Note requires the use of C++0x to eliminate the space between > > in templates.

It does find the route with the minimum number of turns.

#include<iostream>
#include<vector>
#include<string>
#include<set>
#include<memory>
#define D make_pair
#define F first
#define S second
using namespace std;typedef pair<int,int>L;typedef vector<L>R;typedef multiset<pair<float,pair<L,R>>>B;vector<string> M;string l;int z,c,r=0;set<L> s;B b;L n;B::iterator f;R v;void A(int x,int y,int w){n=f->S.F;for(c=1;(z=M[n.S+=y][n.F+=x])==32||(z==75);++c)v.back()=D(w,c),b.insert(D(f->F+c+1./c,D(n,v)));}int main(){for(;getline(cin,l);++r){if((c=l.find(82))!=string::npos)b.insert(D(0,D(D(c,r),R())));M.push_back(l);}while(!b.empty()){f=b.begin();n=f->S.F;v=f->S.S;if(M[n.S][n.F]==75)break;if(s.find(n)==s.end()){s.insert(n);v.push_back(L());A(0,1,83);A(0,-1,78);A(1,0,69);A(-1,0,87);}b.erase(f);}for(c=v.size(),r=0;r<c;++r)n=v[r],printf("%c %d\n",n.F,n.S);}

It Dijkstra's Algorithm for solving the shortest path problem.
To distinguish between multiple equal size routes a long straight line has less weight that multiple short lines (this favors routes with less turns).

Cost of a path:  Len + 1/Len

Looking at Test Case 1:
========================
Thus Path E13 + N2 has a cost of 
      13 + 1/13 + 2 + 1/2
An alternative path E9 + N2 + E4 has a cost of
      9 + 1/9 + 2 + 1/2 + 4 + 1/4

The difference is
      Straight Path:   1/13 <   Bendy Path: (1/9 + 1/4)

In a more readable form:

#include<iostream>
#include<vector>
#include<string>
#include<set>
#include<memory>

using namespace std;
typedef pair<int,int>                   L;
typedef vector<L>                       R;
typedef multiset<pair<float,pair<L,R>>> B;
vector<string>                          M;

string      l;
int         z,c,r=0;
set<L>      s;
B           b;
L           n;
B::iterator f;
R           v;

void A(int x,int y,int w)
{
    n=f->second.first;
    for(c=1;(z=M[n.second+=y][n.first+=x])==32||(z==75);++c)
        v.back()=make_pair(w,c),
        b.insert(make_pair(f->first+c+1./c,make_pair(n,v)));
}

int main()
{
    for(;getline(cin,l);++r)
    {
        if((c=l.find(82))!=string::npos)
            b.insert(make_pair(0,make_pair(make_pair(c,r),R())));
        M.push_back(l);
    }

    while(!b.empty())
    {
        f=b.begin();
        n=f->second.first;
        v=f->second.second;

        if(M[n.second][n.first]==75)
            break;

        if(s.find(n)==s.end())
        {
            s.insert(n);
            v.push_back(L());
            A(0,1,83);
            A(0,-1,78);
            A(1,0,69);
            A(-1,0,87);
        }
        b.erase(f);
    }

    for(c=v.size(),r=0;r<c;++r)
        n=v[r],
        printf("%c %d\n",n.first,n.second);
}

Ruby - 648 characters

Another one that fails the least number of directions test as I can't think of any easy way to embed it in A*.

m=$<.read.gsub /[^RK\n ]/,'#'
l=m.index($/)+1
s=m.index'R'
g=m.index'K'
h=->y{(g%l-y%l).abs+(g/l-y/l).abs}
n=->y{[y+1,y-1,y+l,y-l].reject{|i|m[i]=='#'}}
a=o=[s]
d=Array.new(m.length,9e9)
c,k,f=[],[],[]
d[s]=0
f[s]=h[s]
r=->y,u{u<<y;(y=k[y])?redo:u}
(x=o.min_by{|y|f[y]}
x==g ? (a=r[x,[]].reverse;break):0
o-=[x];c<<x
n[x].map{|y|c&[y]!=[]?0:(t=d[x]+1
o&[y]==[]?(o<<y;b=true):b=t<d[y]
b ? (k[y]=x;d[y]=t;f[y]=t+h[y]):0)})until o==[]
k=a.inject([[],nil]){|k,u|(c=k[1]) ? (k[0]<<(c==u-1?'E':c==u+1?'W':c==u+l ? 'N':'S')) : 0;[k[0],u]}[0].inject(["","",0]){|k,v|k[1]==v ? k[2]+=1 : (k[0]+=k[1]+" #{k[2]}\n";k[1]=v;k[2]=1);k}
puts k[0][3,9e9]+k[1]+" #{k[2]}\n"

c++ -- 681 necessary characters

#include <string>
#include <iostream>
#include <sstream>
using namespace std;
int d[]={-1,0,1,0},i,j,k,l,L=0,p=41,S;string I,m,D("ESWN");
int r(int o,int s){S=s;while((m[o]==32||m[o]==37)&&m[o+S]-35)
if(m[o+S]-p)S+=s;else return o+S;return 0;}
void w(int o){for(i=0;i<m.length();++i)for(j=0;j<4;++j)
if(k=r(o,d[j])){stringstream O;O<<D[j]<<" "<<int((k-o)/d[j])<<"\n"<<I;
I=O.str();if(p-41)--p,m[k]=37,w(k);cout<<I;exit(0);}}
int main(){while(getline(cin,I))m+=I,l=I.length();
d[1]-=d[3]=l;I="";for(i=0;i<m.length();++i)
switch(m[i]){case 82:case 75:m[i]/=2;case 32:break;default:m[i]=35;}
do{for(i=0;i<m.length();++i)if(r(i,-1)+r(i,-l)+r(i,1)+r(i,l))
{if(m[i]==37)w(i);m[i]=p+1;}}while(++p);}

It first replaces all the obstacles on the map with into #s (and changes the values of K and R, to leave headroom in the character space for very long paths. Then it scribbles on the map. An iterative process marks all the successively accessible squares until it is able to reach the kitten in one move. After that is uses the same accessibility checking routine to find a string of positions that lead back to the start in minimal instructions. These instructions are loaded into a string by pre-pending, so that they print in the proper order.

I don't intend to golf it further as it does not properly resolve ties and can not be easily adapted to do so.

It fails on

#####
#   #
# # #
#R#K#
#   #
#####

producing

 $ ./a.out < kitten_4.txt
N 2
E 2
S 2

More or less readable version:

#include <string>
#include <iostream>
#include <sstream>
using namespace std;

int d[]={-1,0,1,0}
  , i, j, k
  , l      /* length of a line on input */
  , L=0    /* length of the whole map */
  , p=41   /* previous count  ( starts 'R'/2 ) */
  , S      /* step accumulator for step function */
  ; 
string I/*nput line, later the instructions*/
  , m/*ap*/
  , D("ESWN"); /* Reversed sence for back tracking the path */

int r/*eachable?*/(int o/*ffset*/, int s/*step*/){
//   cerr << "\tReachable?" 
//        << endl
//        << "\t\tOffset: " << o << " (" << o/5 << ", " << o%5 << ")" 
//        << "  [" << m[o] << "]" << endl
//        << "\t\tStep: " << s 
//        << endl
//        << "\t\tSeeking: " << "[" << char(p) << "]" 
//        << endl
//        << "\t\tWall:    " << "[" << char(35) << "]" 
//        << endl;
  S=s;
  while ( ( (m[o]==32)      /* Current character is a space */ 
        || (m[o]==37) ) /* Current character is a kitten */ 
      && (m[o+S]-35) /* Target character is not a wall */ 
      )
    {
//     cerr << "\t\tTry: " << o+S << "(" << (o+S)/5 << ", " << (o+S)%5 << ")" 
//   << "  [" << m[o+S] << "]" << endl;
    if (m[o+S]-p       /* Target character is not the previous count */ 
    ) {
//       cerr << "\t\twrong " << "  [" << m[o+S] << "] !!!" << endl;
      S+=s;
    }
    else  {             /* Target character *is* the previous count */
//       cerr << "\t\tFound " << "  [" << m[o+S] << "] !!!" << endl;
      return o+S;
    } 
  /* while ends */
      }
  return 0;
}

void w/*on*/(int o/*ffset*/){
//   cerr << "\tWON" << endl
//        << "\t\tOffset: " << o << "(" << o/5 << ", " << o%5 << ")" 
//        << "  [" << m[o] << "]" 
//        << endl
//        << "\t\tSeeking: " << "[" << char(p) << "]" 
//        << endl;
  for(i=0;i<m.length();++i) /* On all map squares */
    for(j=0;j<4;++j)
      if (k=r(o,d[j])) {
//  cerr << "found path segment..." 
//       << (k-o)/d[j] << " in the " << d[j] << " direction." << endl;
    stringstream O;
    O << D[j] 
      << " " 
      << int((k-o)/d[j]) 
      << "\n" 
      << I;
    I = O.str();
//  cerr << I << endl;
    /* test for final solution */
    if (p-41) 
      --p,m[k]=37, w(k); /* recur for further steps */
    cout << I;
    exit(0);
      }
    /* inner for ends */
  /* outer for ends */
}


int main(){
  while(getline(cin,I))
    m+=I,l=I.length();
//   cerr << "Read the map: '" << m << "'." << endl;
  d[1]-=d[3]=l;I="";
//   cerr << "Direction array:    " << D << endl;
//   cerr << "Displacement array: " << d[0] << d[1] << d[2] << d[3] << endl;
//   cerr << "Line length: " << l << endl;
  /* Rewrite the map so that all obstacles are '#' and the start and
     goal are '%' and ')' respectively. Now I can do pathfinding *on*
     the map. */
  for(i=0;i<m.length();++i)
    switch (m[i]) {
    case 82:         /* ASCII 82 == 'R' (41 == ')'  ) */
    case 75:m[i]/=2; /* ASCII 75 == 'K' (37 == '%' ) */
    case ' ':break;
    default: m[i]=35; /* ASCII 35 == '#' */ 
    };
//   cerr << "Re-wrote the map: '" << m << "'." << endl;
  do { /* For each needed count */
//     cerr << "Starting to mark up for step count " 
//   << p-41+1  << " '" << char(p) << "'" << endl;
    for(i=0;i<m.length();++i){ /* On all map squares */
//        cerr << "\tTrying position (" << i/l << ", " << i%l << ")" 
//      << "  [" << m[i] << "]"
//      << endl;
      if ( r(i, -1) /* west  */ +
       r(i, -l) /* north */ +
       r(i,  1) /* east  */ +
       r(i,  l) /* south */ 
       ) {
//    cerr << "Got a hit on : '" << m << "'." << endl;
//    cerr << "\twith '" << char(m[i]) <<" at position " << i << endl;
//    cerr << "target is " << char(37) << endl;
    if(m[i]==37)
      w(i); /* jump into the win routine which never returns */
    m[i]=p+1;
//  cerr << "Marked on map: '" << m << "'." << endl;
      }
    }
  } while(++p);
}