Many theorems and "paradoxes" - Cantor's diagonalization, undecidability of hatling, undeciability of Kolmogorov complexity, Gödel Incompleteness, Chaitin Incompleteness, Russell's paradox, etc. - all have essentially the same proof by diagonalization (note that this is more specific than that they can all be proved by diagonalization; rather, it feels that all of these theorems really use the same diagonalization; for more details see, e.g. Yanofsky, or for a much more brief and less formalized account my answer to this question).
In a comment on the above-mentioned question, Sasho Nikolov pointed out that most of those were special cases of Lawvere's Fixed Point Theorem. If they were all special cases, then this would be a good way to capture the above idea: there really would be one result with one proof (Lawvere's) from which all of the above followed as direct corollaries.
Now, for Gödel Incompleteness and undecidability of the halting problem and their friends, it is well-known that they follow from Lawvere's Fixed Point Theorem (see, e.g., here, here or Yanofsky). But I don't immediately see how to do that for the undecidability of Kolmogorov complexity, despite the fact that the underlying proof is somehow "the same." So:
Is the undecidability of Kolmogorov complexity a quick corollary - requiring no additional diagonalization - of Lawvere's Fixed Point Theorem?