Proving a set is semi-decidable
You are missing a proof that $L_H$ is semi-decidable.

Here are two ways to prove that $S$ is semi-decidable.

First method. The following predicate is decidable: "$M$ reaches state $q$ after running on $x$ for $t$ steps". Hence the following is semi-decidable: "there exist $x,t$ such that $M$ reaches state $q$ after running on $x$ for $t$ steps".

Second method. On input $M$ and $q$, simulate $M$ on all possible inputs (by dovetailing); output $\langle M,q \rangle$ for each state $q$ reached by each of these simulations.