This is why you feel the way you do.
Please start by taking all the guilt and shame away from how you are coping/not coping. No one can function well on constantly broken sleep.
You have my total support and understanding. DS2 is autistic, undiagnosed for years, and never slept. Never ever ever. He woke every 40 mins for at least 30 mins, screaming and thrashing. I was so tired I forgot my own name. My house was a mess. I had to leave a really brilliant job. My self esteem plummeted and I got severe depression. I thought I was a useless mother, useless employee, useless housewife, useless friend, useless wife.
I was just broken with fatigue. It is used as a form of torture for a reason.
Understand that your life is currently limited by your child's sleep patterns. There are things you can't do due to chronic sleep deprivation. With no guilt whatsoever, look into taking time off work with stress while you try to get some medical help for your DC and for you. Maybe a sleep clinic. Maybe melatonin. There was nothing around when DS was small but perhaps there's help now.
For housework I learned to rely on 5 minute 'room rescues.' (Flylady.) Set a timer for 5 minutes and improve a room until the timer sounds. Empty bins, hang up coats and bags, pair shoes, chuck out any junk mail, used food and drink containers etc. Put dirty washing in the machine and turn it on. You will be amazed how much better a room looks after only 5 mins. When I was really tired I used to do it for 2 mins, as half way through 5 mins I'd forget what I was doing, my brain was so shot to pieces.
Every day make a list, if only in your head, but ideally in a notebook, of what you did do, not what you didn't. Very revealing. You might think: Yet again I go to bed with the house a tip, not having changed the sheets etc etc, But when you write it down, you find you spent 3 hours rocking a sobbing baby, 1 hour cleaning up sick, 30 mins trying to get a GP appointment. You realise you did actually prep food, put on a wash load, bathe your child, walk the dog etc.
You also may be ND, as you suggest. Worth trying to get tested for Autism and ADHD - long wait lists but no harm in joining them. And also test for PND. Medication for anxiety and for ADHD if you have it, or for depression will take the raw edge off these days. Taking meds is no sign of failure, It's a sign of self care and finding ways to cope.