Firstly, there is a proof using the motivic spectral sequence (the Atiyah-Hirzerbruch style spectral sequence from motivic cohomology to algebraic $K$-theory). This is written in the master's thesismaster's thesis of Gabe Angelini-Knoll.
Gabe and Andrew Salch are also working to answer this question and a paper is apparently due. From Andrew's website:
"My student Gabe Angelini-Knoll and I have been working on the problem of computing the Waldhausen algebraic K-groups of the algebraic K-theory spectra of certain finite fields. This is an example of "iterated K-theory" and Rognes' redshift conjecture is not known in these cases. Thus far, Gabe and I have (with the aid of a new "THH-May" spectral sequence for computing topological Hochschild homology) computed the homotopy groups of THH(K(F_q)) smashed with the p-primary Smith-Toda complex V(1), for p > 3 and for many (but not all) prime powers q. Gabe is working on the computations of the homotopy groups of the C_p fixed points of this spectrum (this will probably be Gabe's thesis), with the goal of using trace methods to recover the K-groups of K(F_q).
We expect to post and submit our first two papers on this topic before the end of summer 2016. "
