Convex Body with minimum expected l2 norm
If we restrict $K$ and $L$ to be both ellipsoids, then your problem can be solved to any accuracy with an SDP. I know this is not what you asked originally, but it seems we have no solution even for this restricted case, and maybe it can help in general.

So let's say $E$ is the input ellipsoid and we are looking to find an optimal enclosing ellipsoid $J$. There exists a linear map $F$ s.t. $E = FB_2$ and a map $G$ s.t. $J = GB_2$, where $B_2$ is the unit ball. Then $\mathbb{E}_{x \sim J}[\|x\|_2^2] = \frac{1}{n}\mathsf{Tr}(G^TG)$. Also $E  \subseteq J \Leftrightarrow J^\circ \subseteq E^\circ$, where $E^\circ$ is the polar body of $E$. Conveniently, $E^\circ = \{x: x^TF^TFx \leq 1\}$ and $J^\circ = \{x: x^TG^TGx \leq 1\}$. It follows that $J^\circ \subseteq E^\circ$ (and therefore $E  \subseteq J$) if and only if $G^TG - F^TF$ is a positive semidefinite matrix. 

So the SDP is defined by: given a symmetric PSD matrix $M$, find a symmetric PSD matrix $N$ s.t. $N - M$ is PSD and $\mathsf{Tr}(N)$ is minimized. $N$ can be found by solving the SDP and then an SVD will give the axes and axes lengths of $J$.