That's how traditional teleprompters work: screen below reflecting off the half-silvered mirror, camera behind.

Many laptop screens don't open completely flat, so that may not work for this particular concept.

Why not use an iPad. IPad on a flat surface and a webcam or any other camera behind the mirror. You can even build a custom mirror-camera using a raspberry-pi, a rip webcam and a small lcd screen. Sounds like a nice DIY project.