Let $X$ be an Hausdorff space (so that compact sets are closed), and consider the space $F(X)=\{F\subseteq X\mid F\text{ is closed}\}$, with the Fell topology, which has as basis sets of the form $$\{F\in F(X)\mid F\subseteq U_1\land F\cap U_2\neq\varnothing\land\ldots \land F\cap U_n\neq\varnothing\},$$where $U_1$ varies over open sets with compact complement in $X$, while $U_i, i>1$ varies over open sets in $X$.
Since $X$ is Hausdorff we have that $K(X)\subseteq F(X)$, where $K(X)=\{K\subseteq X\mid K\text{ is compact}\}$, and if $X$ is Polish we actually have that $K(X)$ is Borel in $F(X)$ (in fact it is not hard to show that it is $\Pi^0_3$ at worst, but of course it might be simpler, for example for compact $X$). Are there more general conditions ensuring that $K(X)$ is Borel in $F(X)$? If so how complicated can it be as a Borel set in those cases?