Can a nine month old sleep all night? Yes they can, but some do not. Can a nine month old go all night without eating, yes, but some do not. All children are different.
Is this a regular pattern, or has it just started recently? If it is not regular, then you should try to find out if there are any new stress issues or health related issues. Teething is what would first come to mind. Or a digestive problem. Did she check out okay at her last doctor's visit?
My oldest dd had night terrors. She would wake up crying, and nothing I could do would help her except hold her until she went back to sleep. She did sleep all night most nights, except when she had these bad dreams. She was also a sleep walker when she was a young child.
I am not generally a fan of the "crying it out" method, but sometimes it has to be used if the sanity and health of the mother is at stake.
First I would suggest that you try cutting back on day time sleeping, try to have a regular bed time and wake up time each day. Make sure she is eating and drinking enough during the day. Don't give heavy snacks right before bed time. Avoid stimulating play also. Try to find time to cuddle and reassure your daughter a lot during the day.
If all else fails, then letting her cry it out may work. However, I don't believe you should never go and check on her. Something could really be wrong. My dd got out of her crib when she was that age and was lying on the floor. I don't know what would have happened if I had ignored her.
If she is crying for a long time, you could go and check that she is all right. Don't pick her up. Just check, reassure her and then leave. It's up to you how often you check. If she gets used to the idea that you aren't going to pick her up to cuddle or feed her, then she should stop.
You are right that your husband needs his sleep. But so do you. Talk with him about it, ask his advice, and help. If there is no way he will help, can you get someone to come during the day to watch your kids while you nap?
Don't worry about the housework. It will keep. Your kids won't, they are growing up fast.