A number of small, but significant changes
In[1]:= n = 2.48; k = 4.38;(*dn/d[Eta],dk/d[Eta]*)
\[Epsilon] = (n + I k)^2; \[Eta] = -5 10^-3;
d\[Epsilon][An_, Ak_] := 2 (n + I k) (An + I Ak) \[Eta];
rp[\[Theta]_] := (\[Epsilon] Cos[\[Theta]] - Sqrt[\[Epsilon] - Sin[\[Theta]]^2])/(\[Epsilon] Cos[\[Theta]] +
Sqrt[\[Epsilon] - Sin[\[Theta]]^2]);
drp[\[Theta]_, An_, Ak_] := ((d\[Epsilon][An, Ak] Cos[\[Theta]] - d\[Epsilon][An, Ak]/(2 Sqrt[\[Epsilon] -
Sin[\[Theta]]^2])) (\[Epsilon] Cos[\[Theta]] + Sqrt[\[Epsilon] - Sin[\[Theta]]^2]) - (\[Epsilon] Cos[\[Theta]] -
Sqrt[\[Epsilon] - Sin[\[Theta]]^2]) (d\[Epsilon][An, Ak] Cos[\[Theta]] + d\[Epsilon][An, Ak]/(2 Sqrt[\[Epsilon] -
Sin[\[Theta]]^2])))/(\[Epsilon] Cos[\[Theta]] + Sqrt[\[Epsilon] - Sin[\[Theta]]^2])^2;
sol = Simplify[NSolve[{
2 Re[drp[10.0 \[Pi]/180, An, Ak]/rp[10.0 \[Pi]/180]] == 0.001509,
2 Re[drp[60.0 \[Pi]/180, An, Ak]/rp[60.0 \[Pi]/180]] == 0.00287}, {An, Ak}]]
During evaluation of In[1]:= NSolve::svars: Equations may not give solutions for all "solve" variables. >>
During evaluation of In[1]:= NSolve::ratnz: NSolve was unable to solve the system with inexact coefficients. The answer was
obtained by solving a corresponding exact system and numericizing the result. >>
Out[3]= {{Ak -> (-1.48616 - 1.24023 I) - (1. - 1.96183*10^-16 I) Im[An] + (0. + 1. I) Re[An]}}
(*thus it appears*)
Ak == -1.48616 - 1.24023 I + I An