When the battery is nearing the end of its life, it takes more power and time to recharge it. When the tablet is turned on, the energy that charges the battery is close to the energy taken out to run the tablet, hence the slow or no increase in charge percentage.
For a while the battery can be reconditioned by cycling it:
1 - Turn off the tablet and leave it on the charger until the battery light turns green. If the light never goes green the battery is not chargeable.
2 - Unplug the charger and turn on the tablet. Go to System->Interface Setup/Test and record the battery percentage.
3 - Let the battery drain until the tablet turns off.
4 – Repeat from Step 1.
If the recorded percentage level shows a slow increase from cycle to cycle, the battery can be renewed by continuing the above cycles until the percentage does not increase any more.
When the battery becomes un-chargeable, we recommend to take it out permanently and plug the AC-adapter in a battery back-up power supply (UPS). The following unit would be more than sufficient, for about $60: http://www.tripplite.com/en/products/model.cfm?txtModelID=3116