Chosen Solution
MBA 2013 13 inch The connector in question is the one marked 358MA. I’ve tried searching for a replacement connector but as you can tell I have been unsuccessful.
I’m currently using my Mac with just the power cable. Update (02/11/2016) The connector of the battery was cut off and the wires stripped back to align with the solder pads on the logic board. The battery cable was then soldered to the pads of the logic board. Keep in mind the surface mounted connector of the logic board came loose which left the pads exposed.
I had my technician at work hard wire the battery to the board. She cut the connector off the battery, stripped the wire back and soldered it in place. It was done under her microscope. I have to warn you that my issues came about because water was spilled on MRI my mac which damaged the original battery. I got a used one from eBay and tried to solder myself but over heated the connector on the logic board that damaged it. The used battery stopped working recently so I had the technician pull it out and soldered a second used battery on. So far no issues yet.
Well seeing that I have access to a lab that does repair of PCB I asked the technician to remove the female connector from the battery and hard wire it to the board until I can find another 358 MA connector. The battery is charging and all looks well for now. I’m hoping I can find a connector soon.