I'm no expert, but if the egg was broken and didn't come out, she could be seeping the white and yolk, which now, after a few days must be starting to rot. Alternatively (or in addition), the broken shell could have perferated something and she could be infected.
You really need to keep her protected from the flies, and clean her bottom. You should also probably be giving her antibiotics, however, IMO, if you were not able to get the broken egg out, and a vet visit is out of the question, euthanizing her may be the kind thing to do.
Sorry, and good luck.