So this is really an iff I'm trying to prove but the other direction was very straightforward. This is essentially what I'm trying.
Let $\alpha\in M$. Then we want to show that the minimum polynomial of $\alpha$ over $K$ is separable. Let $p(x)$ be this minimum polynomial. Then we can write
$$p(x)=q_1(x)q_2(x)\cdots q_m(x)$$
for some irreducible $q_i\in L[x]$. Then each $q_i$ is separable over $M$, so they all have distinct roots. What I am having trouble showing that this implies the roots of $p(x)$ are all distinct. How can I be sure $q_1$ doesn't have a root in common with $q_3$, for instance?