There are 7 equations in 10 variables. I have assigned a numeric value to three of them, and then found a numerical solution with respect to the others using FindRoot. Unfortunately, k2 has come out negative:
eqa1 = -(\[Kappa]1/2 + I Subscript[\[Omega], c]) a -
2 I \[Lambda] jx a\[Conjugate] -
\[Kappa]2 (2 n a + A a\[Conjugate] - 2 a^2 a\[Conjugate]) == 0;
eqa2 = -(\[Kappa]1/2 - I Subscript[\[Omega], c]) a\[Conjugate] +
2 I \[Lambda] jx a -
\[Kappa]2 (2 n a\[Conjugate] + A\[Conjugate] a -
2 a\[Conjugate]^2 a) == 0;
eqa3 = -(\[Kappa]1 + 2 I Subscript[\[Omega], c]) A -
4 I \[Lambda] jx n -
2 \[Kappa]2 (3 n A - 2 a\[Conjugate] a^3) == 0;
eqa4 = -(\[Kappa]1 - 2 I Subscript[\[Omega], c]) A\[Conjugate] +
4 I \[Lambda] jx n -
2 \[Kappa]2 (3 n A\[Conjugate] - 2 a a\[Conjugate]^3) == 0;
eqa5 = 2 I \[Lambda] jx (A - A\[Conjugate]) - \[Kappa]1 n -
\[Kappa]2 (4 n^2 - 4 a\[Conjugate]^2 a^2 + 2 A\[Conjugate] A) == 0;
eqa6 = 2 Subscript[\[Omega], a] jx -
2 \[Lambda] jz (A + A\[Conjugate]) == 0;
eqa7 = jx^2 + jz^2 - 1 == 0;
eqs = {eqa1, eqa2, eqa3, eqa4, eqa5, eqa6, eqa7};
assigned = {Subscript[\[Omega], a] -> 1,
Subscript[\[Omega], c] -> 1, \[Lambda] -> 1};
startingValues = {{a, 2}, {A, 2}, {jx, 2}, {jz, 2},
{n, 2}, {\[Kappa]1, 2}, {\[Kappa]2, 2}};
sol = FindRoot[eqs /. assigned, startingValues]
Chop[sol, 10^-7]
eqs[[All, 1]] /. sol // Chop
