I'd be very surprised if damage to the screen affected the little charging LED. Although it seems to be showing through a pinhole in the bezel I'm sure that's a separate component.
How to tell whether it's turning on and just the display isn't working? I can think of a few:
1) try to force a restart, by pressing and holding the power button for 20 seconds or so. If it's anything like mine it will vibrate when it restarts.
2) Try to turn it on, give it time to have booted up, then try phoning it - if it rings then it is really on.
3) Try connecting it to a computer. If the computer recognises it then it is responding, which means that something is powered-on.
4) If you left WiFi on, try to turn it on then go to your router's admin screen and see whether it has connected to your network
Just a few examples of ways to check whether it's alive but with a blank screen or simply dead.