Non responsive. Dead battery?

I have an iPad 3 that was missing the digitizer and the LCD was trashed when I got it. Tried charging it and connecting to iTunes but it seems non responsive. No sounds and not recognized by iTunes. It has no home button and the dock connector looks ok though the housing is bent around it. No signs of obvious damage to logic board or batteries.

My question is:

Is it possible that the battery is too dead to take a charge? It's likely been sitting in a drawer like this for months.

Is there a way I can test the dock connector to make sure a charge is passing through to the battery?

Also is there a way to test the logic board to make sure it's still alive.

I've only paid $20 for it but want to make sure it's worthwhile buying a new battery, LCD, and front panel for it.