I'm mostly done with the hardware now, so I'll present the overall design in this post and explain my solutions to some challenges I encountered along the way. There will be another post on software later.
This is back of the mirror, without a cover so far and with some cluttered wiring which I hope to solve through some clips to lead the wires along the frame. The display sits in the center together with its control board, on the top left and right (behind the wooden bars with 2 screws) are the speakers. They are connected to the display as they are the ones that came with it, they are decent enough for speech output and get their signal through HDMI.