After the helpful responses from John and ikku I bought a pair of "HDMI Female Jack 19pin Connectors 180 Degree" from ebay. It was just a few dollars.
I also picked up a new temperature controlled soldering Iron and a ChipQuik low temp solder SMT replacement kit.
The instructions for the kit were to cover the connections with flux, apply the low temp solder, and remove the part. I had a bit of trouble because of the four pieces of the HDMI connector that went through the board for structural support. I ended up clipping those, removing the jack, and then removing the four remaining pices.
Afterwards I cleaned the area with alcohol and then soldering the new jack on. The flux that came with the ChipQuik (seems like really nice stuff, but I don't have a lot of experiance with flux) was very helpful in removing the solder bridges accross the 19-pins on the jack. Afterwards I cleaned again with alcohol.
I booted it and it almost worked... the hdmi was fuzzy and dropping in and out. I hit each of the 19-pins once more with my solding iron and new everything seems to be working well.
- The jack seems like a standard part
- Good flux was key. I think the low temp solder helped, but I might have been able to do this without it. It certainly didn't hurt so I am glad I bought it.
- The temperature controlled solding iron made a big difference. Not sure I could have done this with a cheap iron like I've used before.
Thank you for all the help!