Is "I like how when + phrase" correct?

Yes. In this case how is roughly interchangeable with that, and while maybe a little clunky, the phrase is grammatically correct. In this example there's no clearly superior way to phrase it (that I can think of) other than "I like how..."

EDIT: Here is why I consider how and that roughly interchangeable in the OP's sentence: Simply put, swapping one for the other has no impact on the sentence's logical flow or meaning. Either way, the causal relationship between Katy's question and Lily's reply is clearly established, and that's the role "I like how/that when" is intended to fulfill.

On the surface the phrasing feels redundant. However it falls in line with similar colloquial gems such as "also too" and "I know, right?" (The latter being a debatable case of reverse redundancy). As cumbersome as it is on the ear of a lingual enthusiast, it is a difficult one to argue based solely on grammar. Its widespread use is the single factor that makes it a battle best left unfought.