The lemma is
Let $X_0$, $X$ and $X_1$ be three reflexive Banach spaces with $X_0 \subset X \subset X_1$. Suppose $X_0$ is compactly embedded in $X$ and $X$ is continuously embedded in $X_1$. Let for non-infinite $p$ and $q$ $$W = \{ u \in L^p ([0, T]; X_0) | \dot{u} \in L^q ([0, T]; X_1) \}.$$ Then this compactly embedded in $L^2(0,T;X)$.
I just want to check that I can pick $X=X_1$ here. I have in mind $L^2 \subset H^{-1} \subseteq H^{-1}$.
You can. In fact, the assumption is stronger the smaller $X_1$ is (smaller space has bigger norm, hence it's more difficult to be bounded in that norm). So for the purpose of verifying assumptions one could take some large space for $X_1$ (like $H^{-10}$, if you will) with no penalty in the result. This suggests that there may be a form of the lemma that does not need $X_1$ at all.
The assumption of reflexivity is not needed either, by the way. The sharp (if and only if) form of this lemma was proved by Jacques Simon in much-cited paper Compact sets in the space $L^p(O,T;B)$ which is available from his webpage.