Honestly? I suck it up and deal with it
I've never been as sick as you sound, and I've only got two kids to deal with, but when I'm sick (even if dh IS home) I just get to feel like shit and deal with it. It sucks. Generally, I'll do as little as possible, do dishes and cook, but I'll let the cleaning go. No picking up toys, cleaning of kitchen, etc. I'll lay on the floor and let the boys climb and jump on me. I can lay still and they're entertained with "Jumping Bag Mommy".
Can you farm the kids out to grandparents, in-laws, etc? Even if they're split up (a couple kids go here, a couple go there..) you could at least have a day or two to try and recover.