Try taking it to an AASP (Apple Authorized Service Provider) and have them run a battery test on it to see if it is failed prematurely or if it's just exhausted.

Certification has no meaning unless employed by Apple (as a Genius) or an Apple Authorized Service Provider. The only thing it does it you aren't, is provide a way to differentiate yourself from another candidate should you choose to pursue a job at one of these locations.

Apple's part number for the VSYNC cable for this model is 922-9161, hope this helps. Chris