I would like to prove the following fact, which I learned from a previous MO question.

Let $S_\cdot,T_\cdot\in\mathbf{sSET}$ be simplicial sets, and assume that $T_\cdot$ is Kan. Then there is a weak equivalence $$ |\underline{\mathbf{sSET}} (S_\cdot,T_\cdot)|\simeq \underline{\mathbf{TOP}} (|S_\cdot|,|T_\cdot|) $$

Here is what I have so far: By a Quillen adjunction,
$$
\mathbf{TOP} (|\underline{\mathbf{sSET}} (S_\cdot,T_\cdot)|, \underline{\mathbf{TOP}} (|S_\cdot|,|T_\cdot|)) \cong \mathbf{sSET}(\underline{\mathbf{sSET}} (S_\cdot,T_\cdot),\textrm{Sing}\ \underline{\mathbf{TOP}} (|S_\cdot|,|T_\cdot|))
$$
So we need to find a weak equivalence in the second set. Notice
$$
\textrm{Sing}\ \underline{\mathbf{TOP}} (|S_\cdot|,|T_\cdot|))_k \cong \mathbf{TOP}(\Delta^k\times |S_\cdot|,|T_\cdot|)
$$
And by the universal property of coends, there's a map (*)
$$
|\Delta(k)|\rightarrow \Delta^k
$$

($\Delta(k)$ is the simplicial set corepresenting $[k]\in\mathbb{\Delta}$), so we get a map to $\mathbf{TOP}(|\Delta(k) \times S_\cdot|,|T_\cdot|)$, which is in turn in bijection with $\mathbf{sSET}(\Delta(k) \times S_\cdot,\textrm{Sing} |T_\cdot|)$, and that is the $k^{\textrm{th}}$ space of $\underline{\mathbf{sSET}}(S_\cdot,\textrm{Sing}\ |T_\cdot|)$. So unraveling, we need to find a weak equivalence
$$
\underline{\mathbf{sSET}} (S_\cdot,T_\cdot) \rightarrow \underline{\mathbf{sSET}} (S_\cdot, \textrm{Sing} |T_\cdot|)
$$
We do have the unit map $T_\cdot\rightarrow \textrm{Sing} |T_\cdot|$ of the adjunction in the target, and since all simplicial sets are cofibrant, the result would follow if
this unit map is a trivial fibration when $T_\cdot$ is fibrant (by compatibility of inner-hom with the model structure in $\mathbf{sSET}$). Here's where I'm stuck; it seems like I'm missing a key ingredient to finish.

(*) also here I need to show that these set maps $$ \mathbf{TOP}(\Delta^k\times |S_\cdot|,|T_\cdot|) \rightarrow \mathbf{TOP}(|\Delta(k) \times S_\cdot|,|T_\cdot|) $$ assemble to a weak equivalence of simplicial sets.