Yes, definetely use some rosin paste flux. Simply smear a a small amount onto the entire 1/4" lead and if possible, even get some between the strands but be careful not to bend the strands so much that you start to break them off. Definetely do the pre-tinning Jim describes.
You mention you have an old heavy-duty soldering iron. I would NOT use that here. You don't need that much heat here, and it would be sort of like trying to paint a masterpiece with large magic-markers. Try to locate a soldering pencil, with a tip about the same size as the broken wire or at most, only slightly larger. Maybe it's time to invenst in a nice automatic-temperature-regulating one? Your work will be much more precise, neater, and you'll get the heat right where it is needed most.
When you select the piece of wire you plan for the splice, try to find one that is rather flexible. Until you repot (JB Weld as others suggest) the entire area, the less stress you exert on that little stub of wire the better. Keep the diameter (gage) of the wire small too. At less than 2 amps, you don't need much of a wire. In fact, the existing larger wire is probably only that size for physical strength. I would think an 18 awg will work just fine, but 16 might also be OK.
Good luck, I'm sure you'll do fine!