In some cases a "natural" measure may be a Hausdorff measure (if it exists) that is positive and finite. So we want not just a Polish topology, but a specific metric. Given a metric for $X$ itself, use the Hausdorff metric on $\mathcal{K}(X)$.

In some cases ($X$ countable, with box dimension zero) it may happen that $\mathcal{K}(X)$ has positive, finite Hausdorff dimension. Then use the Hausdorff measure in that dimension.

But typically $\mathcal{K}(X)$ has infinite Hausdorff dimension. There still may be a gauge function $\phi$ so that the Hausdorff measure $\mathcal{H}^\phi$ is positive and finite on $\mathcal{K}(X)$. But it may happen that you always get measure zero or infinity. You can still try to classify $\phi$ into these two classes.

Given a good gauge function for $X$, how is that related to a good gauge function for $\mathcal{K}(X)$?

My student Mark McClure worked out a few of these in his thesis. Also see his references:

McClure, Mark, "The Hausdorff dimension of the hyperspace of compact sets". *Real Anal. Exchange* 22 (1996/97), no. 2, 611–625.

McClure, Mark, "Entropy dimensions of the hyperspace of compact sets". *Real Anal. Exchange* 21 (1995/96), no. 1, 194–202.