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.