iPad 2 battery is not draining completely
So as it turned out, it's a case of a mis-calibrated battery indicator. Here's what I did to re-calibrate it:


Use iPad until it switches off by itself
Charge it back up
After a couple of minutes, it should turn itself back on. Check the battery indicator to see if it's around 2% (as would be expected). If it is, then allow the iPad to be fully charged back up to 100%. If it is not (i.e. it is around 14% as in my experience), proceed to the next step.
Disconnect the iPad from the charger and let it discharge completely again
Repeat from (2).


It took my iPad one cycle of the steps above to get its battery indicator re-calibrated and it has been working fine since. If this doesn't fix your iPad it's probably worth getting AppleCare involved.