I got a new adapter (finally), and it predictably didn't fix anything. I can't find the old battery either so cannot test to isolate if it's the battery or the DC board causing the problem.
Annoyingly, I cannot find a DC board for this model Mac (A1138) in stock anywhere. There isn't even one on eBay at the moment. I can get a new battery for it though.
What are the chances it's the battery refusing to charge and the DC-in board is fine?