I haven't come across a case of a JDM-050 not charging, but it certainly sounds like the charge IC could be bad. With the charge IC being bad you can still connect it and turn it on sometimes even without a cable if you plug a charged battery. It's just the charging circuit inside the chip that could be dead.
The console recognizes the controller so all 4 USB connections are presumed good (5v positive and ground, and both data lines). Does it turn on when you plug a charged battery (without the USB cable plugged in)? If not, you could check the voltage at the battery connector with the battery plugged in. If the voltage dropped, you could be closer to finding the fault. Maybe it's just the capacitor that stabilizes the voltage coming in from the battery. If that's not the issue I would connect the USB cable with the battery still connected and then touch the charge IC (or any chip for that matter) to check if any IC is getting hot.
Another possibility is that a component might have been knocked off the board at some point. A detailed picture would be useful for us to help with that.