It looks to me like this is an adapter that plugs into the HD. Is that correct?
If so, try to contact the manufacturer of the external box and see if you can get the adapter. I doubt you'll have much luck unless you speak Chinese. The next best solution would be to buy a replacement external case, you can get nice ones with a fan for a very reasonable price. I have two Rosewill ext boxes I have been very impressed with.
If it isn't an adapter and does need to be re-soldered, it will need to be done by someone who is experienced with board soldering and limiting the heat transferred beyond the soldering point. I haven't fixed this particular problem, but I have repaired many similar connections.
If you just go after it with a hot-ass soldering iron, you'll get it re-connected, but you will most likely kill one of those little micro-electronics pieces in the picture near the break.
There's a form of soldering heat dam at the link below. It's for plumbers soldering water pipes, but it works the same and gets the point across. (A couple of small pieces of it might well work, but it would be better to get something from an electronics supply house so you know it won't damage the board or components. http://pipefreezekit.com/prodimages/hot-dam-lg.png)
If you look carefully, you will see there is a metal shield/support over the connector. That is soldered to the circuit board for stability of the connector. (I put four red dots on the image to show the soldering spots.) That way, as the drive's power cable is plugged and un-plugged, the tension & force is not on the electrical connections. It looks like it might also connect at the green dot, but I can't tell without turning the board over.
The shield would need to be popped off the connector or otherwise moved out of the way so you can first get to the actual electrical connectors which I marked with blue dots. Those would need to be soldered first, the drive tested, then the shield/support soldered back on.
You don't need to have the drive tested before hand. Either it works or it doesn't, spending money to find out wouldn't make much difference. If it needs to be soldered back on, find someone who is capable and pay them to fix it, or get the tools and supplies and practice on a piece of junk electronics. Go to good will and buy something and unsolder and re-solder components and wiring until you feel like you can fix the drive, then go for it.
You need a soldering iron, (a cheap Harbor Freight 30 watt iron will be fine), some soldering flux, small gauge silver solder (about 1/16" diameter) and some soldering dam for electronics. A lot of guys will say use resin core solder and skip the flux, but I have nothing but problems with resin core solder and prefer to use a water soluble paste flux and silver solder. The newer water-based fluxes are so mild they won't damage boards or components like the older flux would do. Flux is for cleaning and prepping the parts before you solder them.
You put a small amount on one of the pieces to be soldered together, heat it until the flux paste sizzles & liquifies, then touch the solder to the piece and it will melt on. Repeat for the other piece; you have 'tinned' them. Hold them together and touch the soldering iron to them. If you have put enough solder on them during the 'tinning', they will solder together without adding any more solder.
Get the right amount of solder during tinning, and it is the safest way to solder delicate components with the least amount of heat transfer. Use the heat dam material between the soldering point and the component to insure there is no heat transfer.