I try to catch up on my sleep during the day a) but falling asleep for me usually is really hard / b) but my efforts (to fall asleep) are usually of no avail. "of no avail" is formal. The expression comes with a noun or phrase indicating an effort to achieve a goal. e.g. His attempts at lying to his mother were of no avail. She knew what he was doing all along. "I am sleep-deprived most nights." Good. Just add a hyphen.

It is okay. It might be more natural to say: I'm trying to catch up on my sleep because I can't sleep at night. I have sleep deprivation.