Dell Inspiron 1520 shuts down when power supply is disconnected, but battery shows it has a charge

I have an extended life battery purchased from Dell that has seldom been used.  When the battery charge indicator button on the battery is pressed, it shows 5 green LEDs.  If held for 3+ seconds, all lights go out.  As I understand it, this means the battery is charged and is in good repair.  

When I unplug the power supply, the PC immediately shuts down as if there were no battery installed.  What can I do to troubleshoot this and/or make the battery work?