There’s a hidden reset button that might help. I had to use it once myself a long time ago, and it’s been mentioned a few times on the forums though I couldn’t find a post with a picture.
From my experience it seems like letting the battery discharge completely is a bad idea. Something gets confused and the reset button might have to be used even if the battery is fine and can charge. That’s all the more reason to mod the GameShell and put in a larger battery. The one it comes with is lightweight and small, but doesn’t last long on a charge. There are several posts around here discussing battery mods, and it mostly comes down to finding a compatible battery that fits inside the case (unless you want to modify the case too) and also the wires/connectors needed to hook it up.
I’d recommend using @javelinface 's custom OS, or one of the others instead of the official OS too. Out of the box the GameShell was functional, but not great. With a few tweaks like the battery and OS though, it can be made better. 