Couldn't you lie the screen completely flat and build it the other way (reflecting the screen and passing through the camera), avoiding the keyboard problem?
The tutorial is already suggesting harvesting a web cam from a laptop, why not just extract the screen instead? Lots of info and equipment out there for repurposing laptop screens.
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.
Many laptop screens don't open completely flat, so that may not work for this particular concept.