As others have said.. the JMT is well graded with almost no scrambling or particularly difficult terrain. Much of the trail seems more like a back country super highway. Pretty bunch anything that has a moderately aggressive sole with be just fine. The only think that doesn't work are nearly smooth sole tennis court shoes. They are fine on the solid granite, but really dangerous when there is lose material cause they just slide.
I have been very happy using Inov-8 310 Flyrocs along the JMT. They have good traction, light weight, very breathable, dry reasonable quickly if my feet get dunked. The only downside is that the mesh lets in dust which means socks and feet get dirty faster than ideal. Note: I found that switching to flexible sole shoes had me using my calf muscles more than I did with more traditional shoes... this took a bit of time to get used to.
As what would be good for you... sounds like you have a professional to talk with about the knee so I would recommend discussing what's best with them. My personal experience is that my knees got a lot better when I ditched my heavy, relatively inflexible boots and switched to trail runners. I have been even better with ultra flexible soles like inov-8s. My understanding is that the foot system is pretty remarkable, and that the knee and hip take less shock when going barefoot or using a shoe with a flexible sole like the inov-8. That said, switching footwear a few weeks before a major trip brings with it an element of risk.
There are a number of things to help ease pressure on your knees... dump as much pack weight as you can (I found that there was a big difference for *me* if I kept my pack weight below around 30lbs), use poles, a hinged knee brace, being careful to take your decent slow and careful.