Firstly just wanted to say that you're a great mum. It is obviously from your posts.
Secondly, sleep depravation is a type of torture. Humans are designed to sleep. It's perfectly reasonable to not cope with inadequate sleep. Going back in time women would have lived in groups and shared the responsibility of looking after a newborn. Evolution hasn't caught up with the modern world where one person is expected to do this largely on their own as humans still need sleep. I'm not sure if that makes sense, I'm currently sleep deprived and bad enough at expressing myself when I've had adequate sleep. My point is please don't beat yourself up that you're not coping too well at the moment.