There are two boards responsible for battery issues on these machines. Neither is easy to get to. I'd recommend replacing both due to the age of the machine. This should set you back around $50.
The first is the DC input board, as noted. I'd hesitate to name this the culprit myself as you're not seeing any issues with the machine operating off of the adapter, and you're not getting the red circle. Still, it could help. These boards do not require soldering on the Clamshell models, so replacement is easy enough, just not easy to get to the part itself.
The second bit is the charger/trackpad board. This is located right underneath the trackpad and has the battery terminals on it. As you can't seem to charge the battery, this would be my first guess. This part was integrated into the logic board on later machines, thus Clamshells are the only models (IIRC) to have them.
You can find a guide on replacing both at iFixIt, as well as the parts themselves. Good luck!