> Fretful and sleepless nights because he wakes me at night several times are making me a bit exhausted.
Fretful and sleepless nights are making me exhausted because he wakes me several times in the night.
("a bit exhausted" sounds odd, as if you wrote "a bit very tired")
> I have to get p up to 10 times during nighttime.
you can say "I have to get up up to 10 times during the night." but the "up up" can sound a bit awkward.
"I sometimes have to get up 10 times during the night" would convey the same idea
you can also say "10 times a night". It means the same thing, but is maybe a bit more informal.
> I am trying to catch on sleep at daytime but my problem is that I can't fall asleep.
I am trying to catch up on sleep in the day but my problem is that I can't fall asleep.
> And when I am falling asleep he wake up and demands my attention.
And when I am falling asleep he wakes up and demands my attention.