I fix mac computers as a hobby, so I am in no way an expert, but in my limited experience, it sounds like the dc in board needs to be replaced. Replacing the battery will also be necessary. I have two for this model that were third party and they work just fine. Taking the age of the computer into account, it could also be the logic board, but I'd have to get my hands on it to give you a more clear answer. I would start with the cheapest repairs until you find the problem.