I would guess that you are stuck in an awful cycle of feeling so desperate due to sleep deprivation that you have no energy (or the right mental state) to improve the situation.
I have been on my knees due to kid-induced sleep deprivation. Looking back I can see that sleep deprivation was what it was, but at the time I felt uncontrollably angry, tearful, anxious and jittery, and oh so claustrophobic and trapped. I thought I felt those things because I hated being a parent, but no, I feel very different once I've caught up on sleep. Of course my poor child was just as exhausted as me so the tantrums were off the scale and I had no reserves to deal with them well.
Four was the turnaround age for my child, sleepwise, but that's very different from family to family. I would say though, that digging as deep as you can to make the day times as friction free and pleasant as possible sets the groundwork to allow good sleep to come next. It won't make it come, but an equally difficult daytime certainly makes it very hard for a bad sleeper to transition to a good one. Don't stress about tv time or easy food or whatever if those things make the difference between you being a nicer parent to be around, or you not snapping.
It sounds like you've tried a lot of things so don't want ideas, nor do you want to share the specifics of your nights, but feel free to pm me if you would like to hear what eventually helped for me.
It's not going to feel like this forever. It's the lack of sleep making you like you're losing your mind and hating being a mother, it sends you over the edge. Hang in there. Keep talking here, keep getting help wherever you can.
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