The good news is that the broken socket does not look like it is on the actual hard drive.
If you can remove the black plate shown in the picture you should find that the green board is a addon board with a IDE style connector and power connector to the actual hard drive. Hopefully these connections do not have the connector glued in or siliconed as that power connector looks like it is.
A replacement SATA bridge board is a $5-20 item though getting one to fit exactly the same way might be a problem. So you might have to modify the enclosure to fit a new SATA board.
Here's an example part,
http://www.newegg.com/Product/Product.aspx?Item=N82E16812206002&cm_re=sata_bridge-_-12-206-002-_-ProductIt might be time to just get a SATA drive and skip needing a bridge board at all.
To fix this with a soldering iron would require a new SATA socket - the black part the cable fits into. Not a big deal to do but if you have to pay somebody it would likely be less trouble and expense to buy a new SATA bridge board.