A given power might exist as Power[x,2] (which becomes k) or Power[x, -2] (which we want to become k^-1). Therefore we might try the following on the FullForm, since FullForm makes expressions look like lists. The replacements are done for all levels down to Infinity.
(* Mathematica 7 *)
FullForm[(*given equation*)]
Replace[
(* First replacement for all [Sigma]^2 *)
Replace[%, #, Infinity(*Close inner Replace[]*)] &[Power[[Sigma], 2] -> [Kappa]],
(* Second replacement for all [Sigma]^-2 *) #, Infinity] &[
Power[[Sigma], -2] -> [Kappa]^-1 (*Close outer Replace[]*)]