Hi Jim, maybe I can sketch an answer, assuming that I'm not doing something stupid. The question is related to another one on this toy, namely:
By adjunction, your multisimplicial set has as its set of $(n_1,\cdots,n_k)$-simplices the set of continuous maps from $\Delta^{n_{1}}$ to the mapping space $Map(\Delta^{n_{2}} \times \cdots \times \Delta^{n_{k}}, X)$. As $(n_2,\cdots,n_k)$ vary, these spaces form a $(k-1)$-simplicial space, which I'll denote by $Y$. All of the terms of $Y$ are homotopy equivalent to $X$ since the simplices are contractible, and it follows that the geometric realization of $Y$ is homotopy equivalent to $X$. (See the question cited above). Your original $k$-simplicial set is obtained by applying the total singular complex functor $S$ to each space of the $(k-1)$-simplicial space $Y$. Realizing first in the first simplicial coordinate we are seeing the $(k-1)$-simplicial space $|SY|$, and its realization is homeomorphic to the realization you first asked about. Clearly $|SY|$ is equivalent to $Y$, hence its realization is equivalent to $X$.