An OLED screen could solve this, having no light where the screen is black.

OLED have terrible burn in though, they don't like being turned on for long lengths of time.

Someone above suggested adding a motion sensor, so that it would only turn on when you were in front of it.

Hi, I helped hannahmitt make the mirror.

The rectangle of light isn't noticeable at all in normal light if you adjust the brightness on the tablet well. In a dark room it can be noticeable, though.