Oh my love I feel for you, it's awful being so bone tired you can't think straight. I had two non sleepers, both neuro divergent so there's reasons and I'm sure yours isn't the same. But I had 10 years of no sleep and it messes with your health, your head, everything.
Firstly, it's not your failure! Babies don't sleep. Sorry! Your DH needs to stop whining and help more! 9 months is the worst, if she did sleep ok, it'll come again, travel and disruption don't help but it won't last forever.
Can you rest at the weekend? Does she nap? Can you go back to bed then? At that age my children were still in with me my 10 year old is in my bed right now but ignore that but I took the side off the cot and attached it to the bed so I could sort of move him over to give me more space once he was asleep, I breastfed till he was 2.5...it was the only way to sleep once I was back at work when he was 9 months. I went back full time teaching and I can't lie, it was hard, but I slept whenever I could, rested as much as possible, I batch cooked and did easy food, the house went to shit but we survived. I had a 5 year old and a useless husband (now ex) he left it all to me, ignored me when I was laying with my son for hours trying to get him to sleep and trying to settle my 5 year old simultaneously, left everything to me because his job was sooo much more important than mine. Not saying this to compete, but you might want to make sure your DH knows that these years are the hardest for everyone and he needs to stop blaming you, and take on as much as he can to support you as a family. Or he will make things worse, not you. Guilting you won't make it better.
Do you need to drive to work? If so, you have to try to get rest, stay hydrated and eat well to stay well. If not, sleep on the commute? Ask for family and friends to help if you can? Do the bare minimum at work? Enjoy the hot drinks and child free time once you're there at least?
I have a very clear memory of parking up in the street outside my childcare and sleeping in the car for 20 mins before I picked up my daughter, just to make sure I could drive home safe.
Just do what you need to do to get through the first few days, ignore everything else, including your DH if you have to, he might have a demanding job but so do you, two jobs by the sounds of it, his is to make your life easier. If he's getting a full night's sleep then he takes her once she wakes up so you can catch up on sleep.
This too shall pass, remember that, it's not forever.