Infinite sequence has $m$ consecutive digits – regular languages
You don't explain how the input to $L_k$ is encoded, but $L_k$ will be regular under any reasonable encoding.

Indeed, if $w$ contains a run of $m$ many $k$'s, then it also contains a run of $m'$ many $k$'s for any $m' \leq m$. Therefore either $L_k$ is finite (indeed, of the form $\{0,\ldots,m\}$) or it consists of all of $\mathbb{N}$.

You seem to be troubled by the fact that $L_k$ is not necessarily computable from (a description of) $w$. But there is absolutely no such requirement – all we have to show is that for each $w$, the languages $L_k$ are regular. We are not claiming that we can compute $L_k$ from (a description of) $w$.