My initial response was to use "as to" instead of "for why," I'd go with that.

The sentence is grammatically correct. You can swap "for" with "as to" and keep the rest of the sentence exactly the same. However,you need to change it a little bit for regarding: "Director Kim demands an explanation regarding the reason(s) you are constantly missing your deadline." or "Director Kim demands an explanation with regards to why you are constantly missing your deadline." I personally would consider the "as to" sentence the best of all because it sounds very simple and is grammatically pleasing to a stickler for grammar; It is not too wordy. Having said that, they all work grammatically.