eqk1=-((1 + [Nu]) + (1 + [Nu])/12 R kh n) Cn + (n - (1 - [Nu])) Dn == 0;
 eqk2 = (R^2/12 n^2 + (1 + [Nu])/12 R kh n + 2 (1 + [Nu])) Cn - ((1 + [Nu]) (n + 1/12 R kh n^2)) Dn == 0;
 matkh = Normal@CoefficientArrays[{eqk1, eqk2}, {Dn, Cn}];
 khnAll = Solve[Det[matkh[[2]]] == 0, kh][[1]][[1]] // FullSimplify

FullSimplify[Sqrt[Normal[Series[(kh /. khnAll)^2, {n, [Infinity], 2}]]]]