Hi there, that internal battery cable connector have a tiny circuit board attached to the wire leads that goes to the pins of the battery. Now it is possible for the little circuit board to be defective or a wire that is broken inside of it's insulation. If you have your original OS-X disc use disc 1 to run the hardware test and see if it sites that internal battery connection cable. Now do keep in mind that I don't know if that hardware test option on disc #1 will see that battery connection cable as part of the logic board (it's just not known for sure). All you can do is to try it and see what you get otherwise, I would replace that connection battery cable.
Hi, I just saw your response. The apple hardware test from disc 1 may show that the logic board is faulty. If so then you really have no choice but to have it replaced with another one that works properly. You can obtain a recondition logic board through the "We Love Macs" website (www.welovemacs.com), there you should be able to obtain the parts you will need at a price that may be more to your liking and affordability.