[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 could not figure out how to prove the map $X\to\real{Y}$ that Peter May uses in his answer is a weak equivalence in the manner vaguely described in the answer that Peter links to. Unfortunately, the linked answer 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 and not always homotopy invariant. Perhaps there is an easier way. Obviously, Peter's idea to use $Y$ in the first place is *absolutely fulcral*.

Let $X$ be a space. Recall that $Y:\Delta^{\times(k-1)}\to\Top$ from Peter's answer is 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 a map $cX\to Y$ which is objectwise a homotopy equivalence — as Peter remarks — due to the contractibility of the 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 $\Top(\Delta^n,Z)$. Then applying $S$ objectwise to $Y$ and $cX$, we get a map $SY\to S(cX)$ (where $SY$ really means $S\circ Y$, and similarly for $cX$). Since $S$ preserves weak equivalences, then $SY\to S(cX)$ is objectwise a weak equivalence. Thus 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 very simple consequence of the main theorem of the article <a href="http://arxiv.org/abs/1110.1066">*"Reedy categories and the $\Theta$-construction"*</a> by Charles Rezk and Julia Bergner.

now let us 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 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 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.