I thought I had written your post. Honestly I had to think a out whether I had name changed!
I have 4 DC, including a 3 year old and an almost 2 year old. I'm a single parent.
My 2 year old has a similar sleep pattern to yours. He has a sleep disorder and is under a paediatrician for it. Maybe something to look in to? It is soul distroying.
I would cut out the day time nap, I know that means you won't sleep either, but I compensate by going to bed at 8pm. I get up for the day around 3.30am. Which sounds mad but it means I get a bigger chunk of sleep than if I go up at 11pm and try and sleep until 6. I get the bulk of my housework done very early in the morning instead of trying to do it in the evening whilst going back and forth up the stairs to breastfeed my DS back to sleep every half hour. It also eases the frustration of being stuck upstairs when I have things to do downstairs.
(This is one of the times that being a single parent is easier I think, I have no one making demands on my time and can set my own routine)
It also means that after the morning school run is done, I have nothing to feel that I have to get on with, I can sit and play on the internet or drink tea. I don't do baby groups. They bore me to bits. None of my DC have ever been worse off for spending time playing or watching DVDs and I spend time reading and singing songs etc.
The advice I can give about housework and motivation is to get a skip and a load of black bags and just chuck anything and everything you don't use, like or need. I have been known, when everything had got on top of me to chuck my washing up out and buy a cheap replacement dinner service for £5.99 from wilkinsons. Skips are awesome.
You do what you need to to survive.