Two dimensional rook graphs are the Cartesian product of two complete graphs. In this paper we prove that the gonality of these graphs is the expected value of $(n-1)m$ where $n$ is the size of the smaller complete graph and $m$ is the size of the larger. furthermore we compute the 2 and 3 gonalities of these graphs. We also explore the scramble number of these graphs, which is a new graph invariant and a lower bound on the gonality.