I've used a Logitech K480 bluetooth keyboard for a long time to connect my myriad phones and tablets. This one is full sized, without a number keypad, and has a slot for mounting the tablet. No mouse with this keyboard though. The K480 supports 3 distinct devices, selectable with a thumbwheel.

 

pretty much any bluetooth keyboard should work. logitech is always a good brand as far as keyboards go.

 

I have been buying logitech keyboards for more then twenty years,I have been using wireless keyboards that connected with my p.c. and my tablet forever, think it should be all set using it with your Amazon Fire Tablet.