In the last few weeks I have had half a dozen Chromebooks sent to me for repair with a black screen issue. Indeed, when you pressed the power button you could tell that the screen was getting power, but you just saw a black screen and nothing else.
The good news is this was an easy fix. Basically the Chromebook needed a forced reboot. You can restart a Chromebook with the keyboard.
To force reboot a Chromebook press the Power button and the Reload button.