Tylenol is not enough, especially now that that problem has "set in." NSAIDS are usually indicated, because they help with inflammation as well as pain. (NSAIDS are the class of anti-inflammatory, analgesic drugs like ibuprofen, naproxyn, and aspirin.) An antiinflammatory drug can actually address the underlying problem and help you get well, in addition to giving temporary relief of discomfort.
I have a mechanical valve, and I have been able to take NSAIDS under medical supervison. In fact, I took Naproxyn in the hospital for post-surgical pain. It was on my physician-written orders. Some mechanical valve patients even take aspirin, along with coumadin, as their normal anticoagulation regime.
I think it might be time to get another doctor involved -- one who is primarily focussed on your back problems. Physical therapy and chiropractic do very well in patient satisfaction surveys, as treatment modalities for back pain.
I might go to a physiatrist, who is an MD that specializes in physical therapy, and let him or her do the negotiating with the cardiologist about the NSAID. The physiatrist can also prescribe muscle relaxers and PT for you.
Chiropractors can help, but as nonmedical doctors, they cannot prescribe medication, and other MD's do not consider them their equals. I think you need a back doctor who your cardiologist will respect.
Water therapy is great, too. In fact, if you can swim without undue pain, just swimming on your own would have a good chance of helping.