I second the recommendation to take your cat back to the vet for further evaluation.
If there is discharge from both nostrils then it is more likely an upper respiratory infection (viral or bacterial) or allergic problem. If there is discharge just consistently from one nostril then it is very likely there is a problem starting in just that half of the nose. Looking at the discharge, there is almost certainly infection present but coming from one side it is probably secondary to another problem. Possible problems in elderly cat would include growths such as a polyp (which tend to be benign), cancerous growth, or foreign material stuck in the nose. Depending on your location, fungal infections may be considered which can sometimes present as a more unilateral problem.
The other main differential diagnosis – check the teeth! When there is discharge from the nose, it is sometimes easy to assume this is a nose problem, but the roots of the teeth are very close to the nasal cavity. Even if the teeth appear normal from the outside, getting a good set of dental x-rays (under anesthesia obviously) is often a good idea to make sure there are no abscessed roots. A badly infected tooth can progress and drain into the nose, and good news with this in that extraction of the affected tooth will usually solve the problem.
For diagnosis, your cat likely needs some imaging and/or sampling of the nose to be performed. X-rays are usually the most easily accessible and least expensive, but to get good quality skull x-rays sedation is usually needed. To identify more subtle changes advanced imaging such as CT or MRI may be recommended. Rhinoscopy could also be considered in this case (and sometimes a biopsy can be taken this way if needed), although in cats not every practice may do this as it is a small size of endoscope needed.
Please keep us updated on your cat's progress.
