It's hard when we feel like we can't get anything done. With my fourth baby, my husband would ask (nicely), "So what did you do today?" And the response, half the time, was "Grow the baby." To me, good moms hold and feed their babies...sometimes ALL day. Hang in there!
I do have some ideas, too. Do you have any good friends that can come help? With two of my babies, I had a friend (different each time) that would come do my floors or clean my kitchen once a week as they were able, while I sat near by feeding the baby and we could chat. So I was able to socialize, take care of the baby, and was blessed with a fleetingly tidy house. That always cheered me up.
After your baby falls asleep in the sling, try loosening it, and carefully setting your baby into a baby chair (vibrating or basic infant seat) and you carefully back out of the sling. The baby might continue sleeping if she's still snuggled and warm in the sling.
Try making a nest on the couch. I used to put two pillows from our bed on the end of the couch, then I could nestle the baby into the center of the pillows, so he still felt held. (Careful for obstructions with SIDs, etc.)
As soon as your smallest one is a bit older, maybe she will be able to lay on a blanket and watch her siblings play, or they can do puppet shows for her (or whatever) while you tidy. Good luck.