Date: 19/3/2020
So I have ordered the glass (this, if you're interested) which is the right width that the mirror can be portrait and the tv will sit exactly behind most of the glass. It does say it scratches easily, so I will have be very careful not to scratch it too much.
For the frame, I think I can just buy some planks and rout some grooves into in, then slide the glass into the outermost groove and the tv into the innermost. Again this means that there is no need for mountings and the glass and screen will be snug to each other!