When I ignore my H340 for months, the battery runs completely down (not good for the lithium batteries) and the machine is completely unresponsive. When I plug in the AC adapter, I have to let it charge for a good hour before it feels like waking up (if it could drink coffee, I'd try that, but it can't).
If your battery has been dying for a while, you may want to leave it plugged in overnight, then try powering on. Batteries are relatively cheap on eBay, and even though it's a pain, replacing them isn't a horrible experience.
I think there's even a guy selling parts--you could get a new set of buttons & a contact board. The H320 is definitely worth keeping around.