Frame. Mat. Now it's time to mount the Fire ... and this time without the hot glue.
Obviously 3D printed mounting pieces are the way to go here. Each piece holds a corner of the fire, pressing it up against the mat, keeping it the correct distance from the sides of the frame, and being kept in place once the frame back is re-attached. This means that each corner pieces is slightly different so everything lines up "just so" while avoiding buttons and speakers on the Fire itself.
The above photo shows the Fire held in place, with a USB power cable running from the side and out the bottom of the frame. Notice the cut in the frame to allow the power cable out.
This is a testament to my poor woodworking skills. I drilled a hole then used a Dremel to cut down to the hole from the edge. The result was decidedly "meh" but fortunately it won't be particularly visible once the frame in on the wall. Still, I'd like to know the "proper" way to do this.