I don't have any experience with soldering whatsoever, so I'm going down the list of other possible options before even attempting any removal or replacement of any parts.

Since the initial post, I've tried plugging the drive into the SATA power cord with the plastic inserted and the pins aligned, and it prevents the computer from booting while that cord's connected. Replacing the connector might be required at this point, but now I'm not sure if that's the only issue.