Ring structure of Hecke algebra

As is typical in Lie theory, if you want to find an explicit calculation, it's best to start by checking if Macdonald wrote something about it. In this case, you should start by having a look at Spherical functions on a $\mathfrak{p}$-adic Chevalley group, available free here

https://projecteuclid.org/download/pdf_1/euclid.bams/1183529627

This short paper gives an explicit isomorphism (Theorem 1') from the spherical Hecke algebra $L(G,U)$ of compactly supported, continuous, $U$-bi-invariant functions from $G$ to $\mathbf{C}$ to $\mathbf{C}[P^\vee]^W$, where $G$ is a Chevalley group, $U$ is a maximal compact subgroup, $P^\vee$ is the coweight lattice, $W$ is the Weyl group, so by Bourbaki (Chapter 6 of Lie groups and Lie algebras), $\mathbf{C}[P^\vee]^W$ is a polynomial ring on $r=\mathrm{rk}(G)$ generators.

So even though it doesn't directly address your question (only since $\mathrm{GL}_n$ is reductive but not quite semisimple), you will find it easier to read than his book Symmetric functions and Hall polynomials, which contains the relevant material for the general linear group (in which case the spherical Hecke algebra is the algebra of symmetric polynomials).