Sorry you're finding things tough time. I agree with the others about getting out, even if it's just a walk round the supermarket, you might find the baby sleeps for longer while out. My ds always slept better in the pram/sling, I think I went to Sainsbury's every day when he was tiny, and sometimes I could come home, bump the pram up the stairs (we're on the first floor) and leave him sleeping in the hall while I made a cup of tea and chilled out, or even got forty winks. S seems to be the same, as soon as we're out she's asleep - just need to time it so she's got a clean nappy and full tummy. Get yourself dressed and your bag together, then change and feed her and go straight out.
I'm suffering with the lack of sleep too, it really hits me at about 3pm every day for some reason, I suddenly struggle to keep my eyes open! I can't even sleep when the baby does as I've got ds to look after too.
The part of the day I'm finding hardest is now, 4-6ish. The witching hour! S has been cluster feeding at this time, making it really hard to get ds sorted with some dinner (he goes to bed at 7 so too early for us to all eat together) and give him a bath. I've been giving her one boob, starting dinner, another boob and finishing dinner, another one and helping him down from the table, etc etc. But it does mean leaving her crying on her playmat (which she actually seems to like, thankfully, ds hated it but she's quite interested in the dangly animals) for a few minutes while I'm doing things, and ds is having to wait for things.