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I shouldn't worry too much about the battery exploding, they have protection circuitry on the cell to prevent overcharge/overdischarge and short circuit, as well as a charging chip in the eReader.
It sounds like it has been left in a discharged state for a long time and this has raised the internal resistance of the battery, thus it seems to charge quickly but doesn't hold that charge for too long.
I would back up your library via USB and then proceed to open it up.
It is a tricky procedure, there are effectively two bezels, the one you can see and a "smoked acrylic" one that is the light for the IR touch screen. You should aim to try and separate the two and leave the inner bezel attached to the substrate,
The gloop they use to seal and stick it together is horrible and messy - I found Zippo lighter fluid very helpful for softening it and reducing its stickiness.
Go slowly and carefully is the best advice. The front bezel is very thin and can be easily bent out of shape or even snapped if pulled and pushed too hard.
Once you have the front bezel off, the inner bezel, screen and mother board can be removed as one unit via the four screws in each corner.
The interesting stuff is on the back.
Once you have extracted the battery, you should be able to obtain one of similar capacity and dimensions from the usual sources (Ebay/amazon/alibaba etc). If you aren't scared of using a slodering iron, you can reuse the protection circuitry from the old one and save a few pennies.
HTH
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