Yep, that can easily be soldered back together. Slip some heat shrink tubing over one of the wires, strip them back, twist them together, solder, then slip the heat shrink tubing over the solder joint and then use a heat gun to shrink the tubing over the splice.
Then add another layer of black electrical tape to add a little more insulation. Make sure whoever does the soldering knows how to solder.
If you're going to solder them and aren't all that experienced... get the soldering iron heated up. Wipe it off with a cloth then apply solder to its tip so it is bright and shiney. Hold the soldering iron to the BACK of the wires where they are twisted together and apply the solder to the FRONT of the wires. When the wiring is hot enough, the solder will flow into the wires where they are twisted together. Doing it this way assures a good solder joint. Don't just drip solder onto the wires.