$\newcommand{\Psh}{\operatorname{Psh}}
\newcommand{\Sh}{\operatorname{Sh}}
\newcommand{\O}{{\mathcal{O}}}$
Let $X$ be a locale, $\O(X)$ the corresponding frame.

1. What's the localic reflection of $\Psh X$?

We know that
$$
\O(X) \cong \mathrm{Sub}_{\Sh X}(1)
$$
Call $Y = \mathrm{Sub}_{\Psh X}(1)$ the localic reflection of $\Psh X$.
One has $\Psh X \simeq \Sh Y$.

Since $\Sh X$ is a subtopos of $\Psh X$, $X$ should be a sublocale of $Y$, i.e. there should be a nucleus $j$ on $\O(Y)$ such that $\O(X) = \O(Y) / j$.

2. Who's $j$?