No idea, as kindle fire are not in the uk yet, but has it is such a recent product try email the devs at support@athinkingape.com No idea how many people with kindle fires out there to help you.
Guess you figured it out as you have the badge of linked account already. Guess this thread is pointless then.
Same as on Droid: Download and Install KAW on Kindle On Main screen click the little Cogs "Gears" to go into settings. Click Link to Facebook. If you have an account already on that Facebook you will be given the choice to keep your Kindle account or your Facebook Account. Chose the one you want to keep. It will be accessible from both Facebook and from Kindle/Droid The Other Will Be Deleted! Simple as that. If you want to use Mochi instead of Facebook Only the Developers can make that link. eMail Device_Transfers@AThihnkingApe.com to make the request.