I'm sorry to hear you've been having problems since your transplant. I hope that you're at least getting to enjoy some of the benefits of having a new kidney!
As for the weight, different things work for different people as finding your best solution can be really hard. Might be worth trying the low carb, high fat diet, or something like weight watchers. I've found the juice diet has worked for me in the past. It's pretty tough and can be hard to sustain for more than a couple of weeks. You can adapt it slightly so that you can still have a proper meal a day. It does mean eating (or drinking) a lot of fruit and veg, so you will need to keep an eye on your sugars. I'm type 1, so it's relatively easy for me to just adjust my insulin to suit.
With regards to exercise, cycling can be quite tough on your knees (I'm a bit of a cycling enthusiast, so I know all too well!). Few things that might help, get some clips for the pedals so that you can clip your feet to the pedals. This means that you can pull up as well as push down. It means that your using more muscles in your leg, ad this could help to take the strain off your knees. Secondly, and this might sound a little crazy, try turning the resistance on your bike up a little bit. Make sure you're giving your legs something to push against rather than freewheeling. Again, it could help to take the pain away from your knees.
If that doesn't work, try something like swimming. It's much gentler on the joints and is a great 'all over' exercise. Could try walking, but this might hurt your joints too.
Anyway, I know this doesn't answer all your questions but hopefully it helps a little. You said you've already talked to your doctor etc, might be worth trying some private treatments. My mam tried hypnotherapy a few years back. She's not diabetic, but had some weight to lose. She had a few sessions and, log and short of it, taught herself to think we stomach was smaller than it is. I have to be honest, I laughed when she told me she was going to see a hypnotist to shed some pounds, but it's worked for her!
I hope you manage to find a solution that works for you soon.
Take care,
Emma