The following is an image of a proof from Hovey's Model Categories:
How exactly do we know that $s\restriction_{\partial{\Delta[n]}}$ factors through $X_n$?
Since $\partial{\Delta[n]}$ has only finitely many non-degenerate simplices, it is $\lambda$-small for any limit cardinal $\lambda$, but I'm not sure how this (at least directly) implies the factorization we want.
A more rigorous way to state what I said in my comment is to utilize the $(n-1)-$skeleton functor $sk_{n-1}$.
We know that $sk_{n-1} X_n \rightarrow sk_{n-1} L$ is an isomorphism of simplicial sets. We also know that $sk_{n-1} \partial \Delta^n = \partial \Delta^n$.
So any map $s : \partial \Delta^n \rightarrow L$ gives a map $sk_{n-1} s : \partial \Delta^n \rightarrow sk_{n-1} L$ which then gives a map $s': \partial \Delta^n \rightarrow sk_{n-1} X_n$ because $sk_{n-1} L$ and $sk_{n-1} X$ are isomorphic. There is a natural inclusion $i:sk_{n-1} X_n \rightarrow X_n$ which gives you the desired lift $i \circ s': \partial \Delta^n \rightarrow X_n $. Uniqueness of the lift comes from injectivity of $X_n \rightarrow L$ on $(n-1)-$simplices and below.