You may want to check the LCD connector on the board and see if any of those pins are warped or damage.
To confirm this is not a backlight issue, have you attempted to check the LCD under an extremely bright light after pressing the power button? If no picture than this is likely not backlight issue, but there is no harm in confirming.
Sometime, cheap mini replacement digitizers manipulate the backlight or the power of the iPad. If you remove the digitizer, on the back of the home button flex cable there should be two components which control the smart cover on/off feature. They can wreck havoc if missing.
Also, have you attempted unplugging everything, then plugging everything back in, plugging the battery in last? Try that as well.
Let us know!