Kolmogorov complexity of strings in a given language
You haven't explained what is meant by the canonical $n$th word, but I assume that there is an algorithm generating all words in $L$, and the canonical $n$th word is just the $n$th word in this list. One such algorithm could go over all possible words in size-lexicographic order, and check for each word whether it belongs to $L$, outputting the words that are in $L$ in size-lexicographic order.

Given such an algorithm, you can construct a new algorithm which gets $n$ and outputs $x_n$. Using the new algorithm, you can generate $x_n$ by running the new algorithm on the input $n$. Since the new algorithm is fixed, it has some size $C$. A prefix-free encoding of $n$ takes $2\log n$ (in fact, you can do almost twice as good), leading to a description length of $C + 2\log n$. (Depending on your definition of Kolmogorov complexity, you might be able to do away with prefix-freedom, and then you can replace $2\log n$ with $\log n$.)

Note that we have only used the fact that the canonical $n$th word is generated by some algorithm which gets $n$ as input. The particular form of $L$ wasn't important.