I know nothing about the WK. Not knowing what version of radio you pulled out of the JK, I would have to guess that the JK's radio was canbus based, and the WK's may or may not be. There are different speeds of canbus, that could be one problem.
If I were you I would research what kind of radio the WK uses first, then try to see if the JK's radio will work there.
09 BLACK Sahara 4 Door - Auto, G2 4.88s
TF 2.5" Coil Lift w/shocks, TF Frnt Adj Trac, TF FLCA, JKS Discos, JKS Rear Adj Trac, LOD Full Rear GEN2,
A.C.E. Rock Rails, Viper TB: Check the Garage for more.