What I did was move the Kindle app out of the system folder with Titanium Backup (requires root). I may not have needed to do that because my next step was to "backup and uninstall" (again with Titanium Backup). I then went to Google Play/Market and re-installed from there. Now it works fine.
Update: you do have to change Kindle app from a system app to user app if you want to do a "backup/uninstall". Another option may to just to a "force remove", but I wanted to have the app backed up.
You can't fight in here! This is the war room!
Samsung Galaxy S4 (rooted). Kindle Fire HD 7 and 8.9 variants. Both rooted.