Hello,
I'm so sorry to hear about your cat's discomfort. If you have it or can get to any pharmacy or grocery store that has a small pharmacy section, you can apply a bit of oragel onto her gums. It won't solve the problem, but will help numb that area for a short time to give her relief, hopefully until she can get to sleep. Oragel is safe to use. The real problem lies in her dental disease however, and gingivitis, and it is more than likely she'll need a full dentistry by your vet and possible even oral antibiotics to help clear this up. If this is a financial concern, just ask your vet about applying for care credit, which can be a no money up-front, low to no interest payment plan often used when sudden problems like this come up.
Unfortunately, there are no over the counter anti-inflammatory medications which are not toxic to cats, so you have to avoid those (aka do not give any aspirin, tylenol, advil, etc).
Good luck, and don't hesitate to reply if I can help further.
Christine
Certified Veterinary Technician
Licensed veterinary technician (B.S. Mercy College), 10 yrs in animal medicine and training