Are you trying to keep everything spotless all the time?
Accept now it can't be done not now with a baby feeding all night and even less so when you've a crawling toddler to contain.
Below are how me and my OH manage. You have to get your OH to realise housework and caring for the baby isn't your down time and you have to have time to rest and do your own things. If he thnks it can be done let him go for it on his next annual leave, and make that soon.
Personally we have a list of weekly and daily jobs and when the babies were waking at night we slept when they did, if we needed to.
So daily one room is cleaned with a wet wipe and worst corners swept/hovered.(5 min max) Dishwasher loaded/unloaded, bottles made,
Then every few days I do a wash other days batch cook us and baby food (although your own left overs are fine for the most part).
Using things like a wipeable shower cutain under the highchair, wipeable backwards jacket on the baby (and you if needed) for meals, wipeable table cloth, quick wet flannel over baby, table and highchair while they are still in the high chair so nothing dries on etc.
I'd get the cleaner doing your bigger jobs so beds, oven/fridge, and a good clean of a few rooms (depending on time) rather than the basic wash up and wipe round stuff.
Leave all house admin to your OH to do when he gets in or on his lunch break assuming you are looking after the baby in that time (so booking appointments, MOT, online shop, buying presents and cards, etc).