Hi there, you are through to Dr. Jake. Im a UK based vet. Im sorry to hear of your concern with your chicken, Coco.
The signs sound quite concerning, particularly the length of duration - 2 weeks is a long time for a chicken to be sick.
There could be a few things going on. with her having a dirty vent this could be that she has gastrointestinal disease such as a bacterial infection or parasites are also common in chicken which causes them to be lethargic, watery or loose droppings.
the open mouth breathing could be a sign of a respiratory infection or something affecting the respiratory system:
We do see primary respiratory infections in chickens though this often comes with nasal and ocular discharge, coughing or sneezing.
if she isnt laying eggs at the moment there is a possibility she has an egg yolk peritonitis.- this is where the eggs dont go down the reproductive tract and end up in the abdomen setting off and infection which leads to fluid build up. the fluid pushes on the chickens lungs making it difficult for them to breath and presents as lethargy, open mouth breathing, pale wattle and comb
I would recommend making an appointment with a local vet who is happy to see chickens. They will provide a full clinical exam and listen to the lungs and air sacs to check for a respiratory infection, they will also feel the abdomen to check no fluid has built up. They may provide antibiotics and anti inflammatories for this. I would also recommend collecting a faecal sample for them to look at under the microscope to make sure there isnt any signs of parasites. They may also recommend sending a sample off to the laboratory for culture to check for bacterial infections
i hope all this information helps