I have my mirror in a tricky location for WiFi. The mirror does not even have an outlet nearby, so I have a long power-cable running all the way to where my router is. This is actually the closest outlet, but a huge brick wall blocks the WiFi-signal.
To solve the internet connectivity issues I decided to buy a HomePlug. These allow you to send Internet signals through power cables. One HomePlug is connected to my router, and the other one is located in the plugs installed inside my mirror frame. The Raspberry Pi connects with the HomePlug through Ethernet.
This works very well for me, but I have both HomePlugs installed on each side of one power cable. This essentially makes my power cable double as an Ethernet cable. They are supposed to work through different sockets in different rooms as well, but I haven’t tested this myself. In other words you should be able to place one HomePlug by the router, and the other in any socket close to the mirror. The wires in the wall should handle the rest. Good luck 😊