Nutritional deficiency from being allowed to stay broody too long causing neurological symptoms, perhaps..
Vitamin E and Selenium *may* help.
Many common chicken diseases can present as
ataxia and stumbling backwards, including Marek's, New Castle's disease, Infectious Coryza, and more.
Here is a symptom checker tool.. what I like is that it helps me identify things I didn't even know were symptoms. It classifies this walking backwards under "gait".
http://www.poultrydvm.com/views/symptoms.php
Even though broody's eat, dirt bathe, etc.. they are losing condition the entire time and are only built to sustain it for so long. Some will even go into molt. These ladies with excess brooding hormones will literally brood themselves to death waiting for the air to hatch. It's so frustrating, I know!
In addition.. if it happens to be "layer" feed that the broody is eating, then excess calcium intake
long term for birds NOT in lay
. *can* (doesn't mean will) cause gout, kidney failure or even sudden death. There is a genetic component here, but it's a good time to ask what she's been eating routinely including treats and supplements.
You've already spent good money for the vet.. it's often fruitless, but thank you for trying to help her! So now I know you will be trying more things to improve her quality of life.. (which hopefully includes being more persistent about breaking her) I would start with supplementing nutrition.. recommendations for consideration follow, please make YOUR best decision according to what you see, think, feel, have available, etc.
Several product with slightly different formulations but my top choices..
For mature birds..
Rooster Booster (brand)
Poultry Cell or Poultry Booster.. mixed into a small amount of damp feed.. the Poultry Cell doesn't mix well with water in a drinker. I haven't used the other one yet but do like it's overall profile.
For new chicks and as a boost on occasion
Poultry nutri drench.. is an incredible product also that contains amino acids and I've seen instant turn around. Amino acids make a huge difference! Put in a small drinker or administer a drop directly to the beak below the nostrils and the bird will gobble and swallow the liquid as it rolls around into the mouth naturally. (especially useful if birds or chicks are not actively drinking).
Things that may be on hand already and helpful until you get a formulated supplement.. a human B complex crushed and added to feed.. B vitamins are key to all things immune system related including neurological function.. thiamine, riboflavin, and so on.. you can't go wrong and you can NOT overdose B's.
I don't have info on vitamin E dosing, being fat soluble it CAN be overdosed.. I would think breaking a capsule open over feed for up to a week might be no problem, but please try to very accurate dosing if you do go that route.. A link to help, but again use YOUR best judgement please!
https://www.chickenwhisperermagazin...ness/vitamin-supplements-and-when-to-use-them
One quick thought.. standard "supplement" levels while excellent to support immune system health may not be enough to correct an actual deficiency IF there is one. So like even though the nutridrench for example contains some E.. you might need another source of E that doesn't give too much of the other stuff in the formulation to make the correction.
How are her droppings, broody huge and stinky norm.

Can I figure you've already done a thorough parasite investigation.. internal and external.. Did the vet offer a fecal float or have you treated for worms? Sitting hens are prime hosts. Looking at the abdomen and below the vent after dark with a flash light can give the clearest picture for skin. Also is her skin nice looking and normal or angry red? Just a standard line of questions to rule out all possible things that could pose threat to obvious diminished immune system conditions.
Hope she recovers quickly!
ETA: some folks make a chicken chair..
I think the chest goes on the wide part and the bumm kinda hangs over the back end.. otherwise you would need a whole for waste.