The description points to some version of 'vent gleet'. But 'vent gleet' is just a symptom, so we need to determine the actual cause in order to be able to treat her. So- is the cause a fungal (yeast) infection or is it a bacterial infection? An yeast infection can be dispatched easily. If it's bacterial... we can still help her.
A beloved hen of mine went through a yeast infection a few years ago, it is easy to diagnose by smell. I had read about that smell before experiencing it first hand, people were always using unusually strong words to describe it. What can I tell you? The odor is unpleasant and strong, much worse than regular
chicken poop. The solution is a trip to Walmart/etc to buy the house brand of miconazole, which is the standard treatment for women's yeast infections. There will be a moderate dose of embarrassment, lol, but the hen'll get out of trouble. No need to spring for the brand name (Monistat). Then just a
local bath with warm
water and a good application of the miconazole cream in the affected area plus a little inside the vent. After I did this for my hen, a white foam started to come out of her vent, I guess the miconazole started killing the yeast, which was making a fuss before dying. I kept gently wiping her off, then rebathing her after a few hours, and reapplying the cream. Of
course all of this was happening in February, in New England. Your girl too will need to be inside for her treatment.
I generally don't use pharmaceuticals but miconazole is good to have on hand in case you have to deal with sour crop too, which is an yeast infection, or yeast overproliferation it could be called, this time in the crop.
So... Is there a strong(ish) odor around your hen?