Faulty charge up on Dell Inspiron 8500

Hello,

I have an Ispiron 8500 which has been fine until recently...couldn't get the battery to charge..bought new mains power unit..this works well but sometimes even thouh pugged in doesn't always show on icon tray that it is powered up laptop and doesn't appear to be charging battery every time. When it does show and battery appears to be fully charged it lasts for only a short time..maybe 20 mins or so...no lights indicate on the battery pack...should they? Is the battery pack now no good?