detected by computer but won't turn on after being unplugged

I had this exact issue with one I was given by a friend a few years back. The internal voltage of the battery has likely dropped to far and it has gone into “sleep” where it will refuse to accept a charge in order to prevent anything bad from happening (battery swelling/exploding). To fix this guy, you will probably need to get a new battery and install it. Not the easiest thing in the world, but it can be done. iFixit has a guide for removing the battery and logic board, from there would need to de-solder the old battery leads and solder in the new one, then re-assemble.