I wanted to try and find a less nested expression. Here is another (visually cleaner) attempt where f[x] is the original expression on which the operation needs to be implemented.

((FullForm[f[x]] /. Power[#1, #2] -> #3) /. 
    Power[#1, -#2] -> #3^-1) &[(*Variable to be replaced*)x,(*power \
of variable to be replaced*)2,(*The replacing variable*)k]

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[]*)]

I am having a hard time posting what I want to. However my suggestion is to see what the FullForm of the expression is. your sigma^2 may not appear in that form in both places, but your replacement rule only covers the specific pattern that you use

It seems that the image that you tried to upload did not appear. Perhaps post the code?