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. Flux is for cleaning and prepping the parts before you solder them.
The newer water-based fluxes are so mild they won't damage boards or components like the older flux would do. 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. 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. 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. 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. Either it works or it doesn't, spending money to find out wouldn't make much difference. You don't need to have the drive tested before hand. Those would need to be soldered first, the drive tested, then the shield/support soldered back on. 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. It looks like it might also connect at the green dot, but I can't tell without turning the board over. (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. That is soldered to the circuit board for stability of the connector. If you look carefully, you will see there is a metal shield/support over the connector. Here's an annotated close up of your image that I posted: (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. It's for plumbers soldering water pipes, but it works the same and gets the point across. There's a form of soldering heat dam at the link below. 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. I haven't fixed this particular problem, but I have repaired many similar connections. 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 have two Rosewill ext boxes I have been very impressed with. 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 doubt you'll have much luck unless you speak Chinese. If so, try to contact the manufacturer of the external box and see if you can get the adapter. It looks to me like this is an adapter that plugs into the HD.