iPad mini will not turn on
Try this sequentially:


Plug this iPad to the charger and leave it there for an hour or two. Should it be depleted of its battery this will recover it.
Plug the iPad to your computer and attempt to enter the DFU mode. If your device shows up in iTunes in recovery mode and if you are willing to lose all data on it, restore with latest version of iOS and you may put it back into service.
If none above worked, pay a visit to Genius Bar and have the Geniuses have a go at it.