Chosen Solution

As I was disassembling my Mac Mini, it came time to remove the hard drive. While removing the hard drive, I did not disconnect the hard drive’s cable port to the logic board and it ripped entirely off of the logic board.

^ ^ see port here ^ ^ How would I go about reconnecting the port to the logic board?

I’m afraid this is going to require some very specialized tools to do a board level repair and I’m guessing since your asking what to do you don’t have them. You would need magnification–a micro solder station–and some experience in doing this kind of work. I think this may be one case where you should ask Apple about a flat rate depot level repair.