When I cannot get a result from a larger complicated problem I then try a smaller part of the problem

N1=1/4(e*n)(e-1)(n-1);N2=1/4(e*n)(e+1)(n-1);N3=1/4(e*n)(e+1)(n+1);
N4=1/4*(e*n)*(e-1)*(n+1);N5=1/2(n)*(1-e^2)*(n-1);N6=1/2*(e)*(e+1)*(1-n^2);
N7=1/2*(1-e^2)*(n+1)*(n);N8=1/2*(e)*(e-1)*(1-n^2);N9=(1-e^2)*(1+n^2);
x1=0;y1=0;x2=2;y2=0;x3=2;y3=1;x4=0;y4=1;x5=1;y5=0;x6=2;y6=1/2;
x7=1;y7=1;x8=0;y8=1/2;x9=1;y9=1/2;
x=Simplify[N1*x1+N2*x2+N3*x3+N4*x4+N5*x5+N6*x6+N7*x7+N8*x8+N9*x9];
y=Simplify[N1*y1+N2*y2+N3*y3+N4*y4+N5*y5+N6*y6+N7*y7+N8*y8+N9*y9];
j=Det[{{D[x,e],D[x,n]},{D[y,e],D[y,n]}}];
M1=Simplify[1/j{{D[y,n],-D[y,e]},{-D[x,n],D[x,e]}}.{D[N1,e],D[N1,n]}];
(*K1=Integrate[Integrate[(M1[[1]]*M1[[1]]+M1[[2]]*M1[[2]])(j),{e,-1,1}],{n,-1,1}]*)
integrand=Apart[Simplify[(M1[[1]]*M1[[1]]+M1[[2]]*M1[[2]])(j)]]
Map[Integrate[Integrate[#,{e,-1,1}],{n,-1,1}]&,integrand]

The full integral is very slow so I tried taking the large complicated rational polynomial Apart into a sum of smaller rationals. I cannot guarantee that is valid when I then integrate that result one part at a time, but when I do that it repeatedly complains that some of those integrals do not converge.

And if I look at

Plot3D[M1,{e,-1,1},{n,-1,1}]

and

Plot3D[integrand,{e,-1,1},{n,-1,1}]

that gives me some idea why trying to integrate over that region may be difficult.

Are you certain that your problem is well defined?

Please check all this very carefully to try to make certain that I have not made any mistakes in transforming your problem.