If you're experiencing a battery that drains fast or doesn't charge, make sure the battery is left to fully charge overnight. When plugged in the LED will turn red and turn off when fully charged. It should take about 2-4 hours to achieve a full charge.
Does the unit power on with the AC adapter? This unit bypasses the battery when plugged in.
Is the unit being powered off/shut down? Keep in mind it still uses power while in sleep mode. Power off vs Sleep mode: