Hey Liberty,
I trained in TKD (ITF) for several years and *loved* it, it was a great experience and taught me a lot in terms of confidence and self-defense. I also gained a lot of upper body & abdominal strength. I wasn't cycling then so can't tell you if they make a good balance, but I did get injured *a lot* in TKD, broken ribs and fingers, lots of joint and tendon issues. This was almost all from sparring. I think patterns, stretches, line exercises would all be excellent, and a good complement to cycling, but I would caution you against joining a group that does a lot of sparring if injuries are getting you down with cycling...
Anne