Oh OP, I feel for you, I really do. I was also solo from the day I took my DD home. She had a cows milk allergy that took 10 days to diagnose (and longer to get under control) plus some breastfeeding difficulties etc. So it was hell. I essentially turned nocturnal. I found the nights SO hard but somehow only when I tried to sleep.
Essentially, in the day I followed her wake windows as my own. Up for an hour and a half, sleeping for a bit anf so on. Then at night, I'd get myself set up on the sofa, put on reruns of police interceptors, and stayed up. This sounds mad but dealing with sleep deprivation through the day just felt more manageable for me..
My mum came over one day while I was at breaking point and suggested co sleeping. We did the whole safe sleep, I curled around her in a C shape and honestly thought I was so tired I'd conk.
I did not. We only started co-sleeping around age 1.5 when she was poorly. At 4, she's now regularly in my bed!!
For now it's survival. If your mum can hold her for a couple hours, take that time to be horizontal. Focus on resting as much as sleeping. Being all snuggled up is a good halfway.
You've got this. It won't always be this hard, I promise.
Have a chat to HV too if needed, for us it was silent reflux so actually once she got treatment she was much better laying, until then I needed to sit her up for up to an hour after a feed. Which, when they're taking 40 mins to feed, feeding every two hours and needing to be sat up for an hour, left little in the way of sleep!