[This answer is mostly a very long comment to Peter's answer.] $\newcommand{\real}[1]{\lvert #1\rvert} \newcommand{\Map}{\operatorname{Map}} \newcommand{\Top}{\mathrm{Top}} \newcommand{\sSet}{\mathrm{sSet}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\To}{\longrightarrow}$
I was unable to prove the map $X\to\real{Y}$ that Peter May uses in his answer is a weak equivalence. Unfortunately, the answer Peter links to gives no details. The result would be easy if $Y$ were Reedy cofibrant, but it does not seem to be. [Perhaps someone will correct me here.]
So here is an alternative method which proves directly that the map $\real{SY}\to X$ is a weak equivalence, thus answering the question. It avoids any use of realizations of (multi) simplicial spaces, which are problematic by not always being homotopy invariant. Perhaps there is an easier method. In any case, Peter's idea to use the multi-simplicial space $Y$ in the first place is certainly absolutely fulcral.
Let $X$ be a space. Recall that the object $Y:\Delta^{\times(k-1)}\to\Top$ from Peter's answer is a multi-simplicial space whose space of $(n_2,\ldots,n_k)$-simplices is the space of maps $\Map(\Delta^{n_2}\times\ldots\times\Delta^{n_k},X)$. Let $cX$ be the constant $(k-1)$-fold multi-simplicial space with value $X$. Then we have an obvious map $cX\to Y$ which is objectwise a homotopy equivalence — as Peter remarks — due to the contractibility of products of simplices.
Let $S:\Top\to\sSet$ be the singular complex functor taking a space $Z$ to the simplicial set whose set of $n$-simplices is the set of maps $\Top(\Delta^n,Z)$. Applying $S$ objectwise to $Y$ and $cX$, we get a map $SY\to S(cX)$ (where $SY$ actually stands for $S\circ Y$, and similarly for $cX$). Since $S$ preserves weak equivalences, then $SY\to S(cX)$ is objectwise a weak equivalence. Now we conclude the map on realizations $\real{SY}\to\real{S(cX)}$ is an equivalence. This follows from the fact that any functor $\Delta^{\times(k-1)}\to\sSet$ is Reedy cofibrant: this is a simple generalization of proposition 15.8.7 of Hirschhorn; alternatively, it is a particular case of the main theorem of the article "Reedy categories and the $\Theta$-construction" by Charles Rezk and Julia Bergner.
Let us now view functors $\Delta^{\times(k-1)}\to\sSet$ as $k$-fold multisimplicial sets instead. As Peter observes, $SY$ is then just the $k$-fold multisimplicial set described in the question. So it suffices to prove that the natural map $\real{SY}\to X$ is a weak equivalence. But we already saw above that $\real{SY}\to\real{S(cX)}$ is a weak equivalence. Moreover, $\real{S(cX)}=\real{\diag(S(cX))}=\real{SX}$ as Tom Goodwillie observes in a comment above (and the simple fact that $\diag(S(cX))=SX$). But $\real{SX}$ is well-known to be weakly equivalent to $X$ (the case $k=1$ of the question). Thus the map $\real{SY}\to X$ is the composite of weak equivalences $$ \real{SY}\To\real{S(cX)}=\real{SX}\To X $$ and is itself a weak equivalence.