I finished the Kairos dragon and he was great for a week. Then, he froze in one location on the screen and I have been unable to move him for weeks. Any ideas on how to get him going?
This sounds like a pretty straightforward bug, though not one I've heard of. You can use his "Summon Dragon" ability as DKnight points out - however that is only really relevant for him flying around. If you are having problems using his ability, I would consider that more serious.
As others point out, on iOS your DragonVale account is tied to your GameCenter login, so you can safely delete the app and re-install (just read the prompts carefully when doing so, to retain data). I presume the same option exists for Android devices, since most of your park is stored "in the cloud" - but I'm not an expert on Android. If you delete and reinstall your app, it will likely "reset" your park to a known good state, hopefully with a fixed Kairos.
Finally, check frequently for app updates. This app gets updated all the time, and may of those updates include bug fixes. Possibly this bug has been addressed in newer versions. I know they had a lot of problems with Kairos initially.