Have you tried turning it on with only the battery, not with the charger?
You can also try this thread -
Solved! - Acer laptop not switching ON; even No charging light | Tom's Guide Forum (tomsguide.com)
Even though he had no charging light, I think you should add the extra step that webworkings mentioned after removing the battery and charger- Press and hold the power button for 30 seconds before putting the battery back in.
And if that didn't work, try using a different adapter. Maybe the battery is empty, and the adapter isn't providing enough power. Or since the LED indicator is on, maybe it is charging, but very slowly- If you don't have another adapter to try it with, try leaving the laptop plugged in for a while to...