Savedbygrace's comment made me wonder about whether cats really do have bird killing properties, and turns out they really do! Pasteurella septicaemia. From what I've can tell, it's the avian strain of pasteurella multocida infection (might be wrong, I'm not a vet myself) that comes from the bacteria in a cat's mouth.
Cats have certain bacteria in their saliva that can kill/cause infection in other animals, and in birds, this is called Pasteurella septicaemia. Because it's in their saliva, the bird doesn't even need to get bitten; if the cat got spit anywhere on the bird, that bird might get sick from preening its feathers afterwards. This seems to match up with what you said about cats being around but not finding any injuries.
Unfortunately, most birds will die from the infection within 24-48 hours. The best chance you have is getting her to a vet quickly, at least as far as I know, and there seem to be a variety of antibiotics that can be used to help her.
I got my information mostly from here:
http://www.pigeon-aid.org.uk/pa/html/bite_wounds.php
https://www.thriftyfun.com/Bacteria-From-Cat-Bites-Can-Be-Deadly-to-Birds.html
I can't say for sure this is what your hen has, but it's possible. I hope she gets better, but I'd be prepared for the worse.
Just as everyone else has said, you've done a great job looking after her, and if the worse happens it's not your fault. You can't save them all.