Oh OP! My child only just started sleeping properly very recently and they are nearly 4! I completely understand about the whole "I'm so sleep deprived I wish I was dead" feeling.
These are some tricks I picked up, I hope some of them help you?
Go to bed as soon as your partner gets home from work. Put ear plugs in.
When your child wakes up at around 11, leave this wake up to your partner too!
Ditto any wake ups after 5am, partner can deal with those. Make him help you.
Grave yard shift 11pm - 5am to deal with now. If you can have a couple of nights sleeping 6-11 and 5 - when your partner leaves for work you will feel so much better!
Things only really improved for us when we stopped cosleeping and enforced child sleeping in own cot. We also cut out day time naps for child. Final step ( and I think this is what cured my child's sleep problems at 3.5), was to tell child we had ran out of milk when they woke up in the night, so they only had water. Now they doesn't bother waking up at all. I also did rapid return to bed eg if child got out of bed (when they moved to a toddler bed that is), I took their hand, said nothing at all, and walked them back to bed.
Let me know if you think any of these ideas are feasible! If not I will have a rethink.