I have something similar happend to me!
This is a discussion on Leave iPad on for a long time. within the iPad 1 Forum forums, part of the Apple iPad Discussions category; I have something similar happend to me!...
I have the iPad repaired now. While the repair person had the iPad open to took pictures of the inside. I did not realize how big the batteries are.
I kept the broke part. Turned out not to be the outer glass. It was the glass under the top glass.
Sent from my IPAD 1st generation. WIFI only. 64GB