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Let $\textbf{Top}$ be the category of all topological spaces. I mean to include the bad ones – topological spaces not locally contractible, topological spaces not having the homotopy type of a CW-complex, non-discrete totally disconnected spaces, non-$T_0$ spaces, you name it. As is well known, $\textbf{Top}$ is not cartesian closed, but this is no obstruction to getting a decent simplicial enrichment:

1. Given topological spaces $X$ and $Y$, we have a natural simplicial set $\textrm{Map} (X, Y)$ where the $n$-simplices are the continuous maps $X \times \left| \Delta^n \right| \to Y$. From this definition it is immediate that $\textrm{Map} (X, -) : \textbf{Top} \to \textbf{sSet}$ preserves limits and therefore has a left adjoint, which I write as $X \odot {-} : \textbf{sSet} \to \textbf{Top}$.

2. Since $\left| \Delta^n \right|$ is compact Hausdorff, it is exponentiable a fortiori, so $\textrm{Map} (X, Y)_n$ is equivalently the set of continuous maps $X \to Y^{\left| \Delta^n \right|}$. Hence $\textrm{Map} (-, Y) : \textbf{Top}^\textrm{op} \to \textbf{sSet}$ also preserves limits and has a left adjoint, which I write as ${-} \pitchfork Y : \textbf{sSet} \to \textbf{Top}^\textrm{op}$.

   (This probably surprises some people, since $\left| K \right|$ is not necessarily exponentiable for an infinite simplicial set $K$. The short answer is that $X \odot K$ is not necessarily the same as $X \times \left| K \right|$, and $K \pitchfork Y$ is not necessarily the same as $Y^{\left| K \right|}$. Rather, $X \odot K = \int^n X \times \left| \Delta^n \right| \times K_n$ and $K \pitchfork Y = \int_n Y^{\left| \Delta^n \right| \times K_n}$. When $X$ (!!) is exponentiable then $X \odot K \cong X \times \left| K \right|$.)

3. There is an evident simplicial composition map $\textrm{Map} (Y, Z) \times \textrm{Map} (X, Y) \to \textrm{Map} (X, Z)$: given $g : Y \times \left| \Delta^n \right| \to Z$ and $f : X \times \left| \Delta^n \right| \to Y$, define $g \circ_n f$ to be the composite $$\require{AMScd} \begin{CD} X \times \left| \Delta^n \right| @>{\textrm{id}_X \times \delta_{\left| \Delta^n \right|}}>> X \times \left| \Delta^n \right| \times \left| \Delta^n \right| @>{f \times \textrm{id}_{\left| \Delta^n \right|}}>> Y \times \left| \Delta^n \right| @>{g}>> Z \end{CD}$$ where $\delta_{\left| \Delta^n \right|} : \left| \Delta^n \right| \to \left| \Delta^n \right| \times \left| \Delta^n \right|$ is the diagonal embedding.

   (Basically we are exploiting the fact that every object – so $\left| \Delta^n \right|$ in particular – in a cartesian monoidal category has a unique comonoid structure. You can do this kind of thing whenever you have a comonoid in a monoidal category.)

4. We get an $\textbf{sSet}$-enrichment of $\textbf{Top}$, but unfortunately it is not clear whether it it tensored and cotensored. I have seen it claimed that we have tensors and cotensors for finite simplicial sets, but there is some point-set topology involved in checking this. I think finite tensors and cotensors suffice, at any rate.

   (If we restrict to a complete and cocomplete cartesian closed subcategory of $\textbf{Top}$ containing the standard simplices and their finitary products, then there is no problem, and the subtlety about $\odot$ and $\pitchfork$ also goes away.)

5. In fact, $\textrm{Map} (X, Y)$ is a Kan complex for all $X$ and $Y$: indeed, the natural map $$\textrm{Hom}_\textbf{sSet} (\Delta^n, \textrm{Map} (X, Y)) \to \textrm{Hom}_\textbf{sSet} (\Lambda^n_k, \textrm{Map} (X, Y))$$ is a split surjection, because it can be naturally identified with the natural map $$\textrm{Hom}_\textbf{Top} (X, Y^{\left| \Delta^n \right|}) \to \textrm{Hom}_\textbf{Top} (X, Y^{\left| \Lambda^n_k \right|})$$ induced by the inclusion $\left| \Lambda^n_k \right| \to \left| \Delta^n \right|$, which has a (non-canonical) retraction.

Let $\textbf{Top}$ be the category of all topological spaces. I mean to include the bad ones – topological spaces not locally contractible, topological spaces not having the homotopy type of a CW-complex, non-discrete totally disconnected spaces, non-$T_0$ spaces, you name it. As is well known, $\textbf{Top}$ is not cartesian closed, but this is no obstruction to getting a very good simplicial enrichment:

1. Given topological spaces $X$ and $Y$, we have a natural simplicial set $\textrm{Map} (X, Y)$ where the $n$-simplices are the continuous maps $X \times \left| \Delta^n \right| \to Y$. From this definition it is immediate that $\textrm{Map} (X, -) : \textbf{Top} \to \textbf{sSet}$ preserves limits and therefore has a left adjoint, which I write as $X \odot {-} : \textbf{sSet} \to \textbf{Top}$.

2. Since $\left| \Delta^n \right|$ is compact Hausdorff, it is exponentiable a fortiori, so $\textrm{Map} (X, Y)_n$ is equivalently the set of continuous maps $X \to Y^{\left| \Delta^n \right|}$. Hence $\textrm{Map} (-, Y) : \textbf{Top}^\textrm{op} \to \textbf{sSet}$ also preserves limits and has a left adjoint, which I write as ${-} \pitchfork Y : \textbf{sSet} \to \textbf{Top}^\textrm{op}$.

   (This probably surprises some people, since $\left| K \right|$ is not necessarily exponentiable for an infinite simplicial set $K$. The short answer is that $X \odot K$ is not necessarily the same as $X \times \left| K \right|$, and $K \pitchfork Y$ is not necessarily the same as $Y^{\left| K \right|}$. Rather, $X \odot K = \int^n X \times \left| \Delta^n \right| \times K_n$ and $K \pitchfork Y = \int_n Y^{\left| \Delta^n \right| \times K_n}$. When $X$ (!!) is exponentiable then $X \odot K \cong X \times \left| K \right|$.)

3. There is an evident simplicial composition map $\textrm{Map} (Y, Z) \times \textrm{Map} (X, Y) \to \textrm{Map} (X, Z)$: given $g : Y \times \left| \Delta^n \right| \to Z$ and $f : X \times \left| \Delta^n \right| \to Y$, define $g \circ_n f$ to be the composite $$\require{AMScd} \begin{CD} X \times \left| \Delta^n \right| @>{\textrm{id}_X \times \delta_{\left| \Delta^n \right|}}>> X \times \left| \Delta^n \right| \times \left| \Delta^n \right| @>{f \times \textrm{id}_{\left| \Delta^n \right|}}>> Y \times \left| \Delta^n \right| @>{g}>> Z \end{CD}$$ where $\delta_{\left| \Delta^n \right|} : \left| \Delta^n \right| \to \left| \Delta^n \right| \times \left| \Delta^n \right|$ is the diagonal embedding.

   (Basically we are exploiting the fact that every object – so $\left| \Delta^n \right|$ in particular – in a cartesian monoidal category has a unique comonoid structure. You can do this kind of thing whenever you have a comonoid in a monoidal category.)

4. We get an $\textbf{sSet}$-enrichment of $\textbf{Top}$. I have convinced myself that it is tensored and cotensored (but watch this other question).

   (If we restrict to a complete and cocomplete cartesian closed subcategory of $\textbf{Top}$ containing the standard simplices and their finitary products, then there is definitely no problem, and the subtlety about $\odot$ and $\pitchfork$ also goes away.)

5. In fact, $\textrm{Map} (X, Y)$ is a Kan complex for all $X$ and $Y$: indeed, the natural map $$\textrm{Hom}_\textbf{sSet} (\Delta^n, \textrm{Map} (X, Y)) \to \textrm{Hom}_\textbf{sSet} (\Lambda^n_k, \textrm{Map} (X, Y))$$ is a split surjection, because it can be naturally identified with the natural map $$\textrm{Hom}_\textbf{Top} (X, Y^{\left| \Delta^n \right|}) \to \textrm{Hom}_\textbf{Top} (X, Y^{\left| \Lambda^n_k \right|})$$ induced by the inclusion $\left| \Lambda^n_k \right| \to \left| \Delta^n \right|$, which has a (non-canonical) retraction.

Question 2. Does the coreflector $\mathcal{T} \to \mathcal{S}$ have any good properties? I believe it preserves coproducts.

Question 2. Does the embedding $\mathcal{S} \to \mathcal{T}$ have any limit preservation properties? Are finite products preserved? (I think finite products are preserved if we construct $\mathcal{T}$ using a convenient category of topological spaces instead. This must have something to do with whether geometric realisation preserves finite products.)

[Edit: Corrected some false claims.]

Observation 2. $\mathcal{S}$ is a reflective localisation of $\mathcal{T}$. This would be a consequence of general theory (because the Quillen model structure is a left Bousfield localisation of the Hurewicz model structure) if the answer to question 1 is yes, but I think the usual construction of the Quillen model structure already contains a direct proof of this fact.

Question 2. Does the reflector $\mathcal{T} \to \mathcal{S}$ preserve finite products? What about finite limits in general?

[Edit: Corrected some false claims and modified questions accordingly.]

Observation 2. $\mathcal{S}$ is a coreflective localisation of $\mathcal{T}$. This would be a consequence of general theory (because the mixed model structure is a right Bousfield localisation of the Hurewicz model structure) if the answer to question 1 is yes, but I think the usual construction of the Quillen model structure already contains a direct proof of this fact.

Question 2. Does the coreflector $\mathcal{T} \to \mathcal{S}$ have any good properties? I believe it preserves coproducts.