In the paper of Casselman and Shalika they give an explicit formula for the spherical Whittaker function of an unramified principal series. Apparently, upon combining their formula with the Weyl character formula, one can get an evaluation of the Whittaker function in terms of the Satake parameter of the representation. Can someone please explain how to do this, or give me a reference where it is done?
I think something very close to what is asked in the original question is explained by Haines, Kottwitz and A. Prasad in their review article.
A partial reference is http://sporadic.stanford.edu/bump/rallis.ps. This is a survey article but the discussion is perhaps complete enough to fill in the details for $GL_n$. The general case of an arbitrary split reductive group is also discussed but less completely (there's surely a more comprehensive treatment of this somewhere in the literature, but I'm afraid I don't know where). In either case you might want to supplement these with a discussion of the Satake isomorphism itself. One reference I know for this is Section 2.3 of Gelbart/Shahidi's book Analytic properties of automorphic $L$-functions.