OP,
It sounds like you are in hell.
An extended period of little sleep is horrendous.
Unfortunately I do think you have to get A LOT tougher.
You need to make it very clear to the 5 year old that mummy is not getting enough sleep and is very tired.
There is no way I would entertain for a minute a child coming in for a chat in the middle of the night.
Utterly ridiculous.
Next time it happens you need to be cross and send her straight back to bed.
Absolutely no chat, no smiles, nothing. Straight back to bed.
Let the 2 year old get sick.
Clean her up and again ignore and straight back into bed.
Sleep is crucial.
You need to protect you mental health at all costs.
You will become ill on so little.
Your family need you well.
If that means you are tough at night, so be it.
5 year old at school.
1 hour is more than enough.
Take a few days off if it helps you.
Don't worry about the house and be kind to yourself.
You are doing a great job