If the battery pack is NiMH or NiCd, which is likely for an older device, it is 6V wired in series (1.2V each rather than 1.5V for alkaline).
So the 12V power supply likely just charges the 6V battery pack via a step-down converter/regulator in the amp.
It seems most likely that the rechargeable batteries are spent and need to be replaced. When replaced, they will charge, green light will come on when done charging, and device will work since it needs functional batteries even when plugged in to the charger.
Other things could be wrong, but getting a new battery pack would likely be part of, if not all of, the solution.