The plot thickens
Yes, I did some more googling and you want to use borax, not boraxo soap.
And here is another description of mending resistance wire with borax. An added dimension is that brass wire is used to strengthen the joint.
Method: Repairing resistance wire
Why
Heating elements are found in many places - hair dryers, soldering irons, clothes irons, and many many more. They also have a limited lifetime and tend to break. Somehow, sometime during the operation a part of the wire gets locally weakened and its resistance raises. It may be because of a surface contamination that causes localized changes in the composition of the wire, mechanical damage lowering its cross-section, or locally diminished heat removal. The resistance of that spot raises in comparison with the rest of the wire, and more power gets released there due to the Ohm's law. More power means higher temperature, higher temperature means increase of the resistance, and increase of the resistance means higher locally released power; and soon we have a nice runaway. Then the device stops working, its heater staying cool like if it stars in a Coca Cola ad.
Such device more often than not gets thrown out, even if there is a plenty of life still hidden in the rest of its construction.
How
The heating elements are usually made of a wire, coiled in a suitable shape on a refractory support (ceramics, mica, etc...), composed of a material with high electrical resistivity and decent resistance to heat. The most common such materials are nichrome (alloy of nickel, chromium, and sometimes iron) and kanthal (alloy of iron, chromium and aluminium).
The wires are coated with a layer of oxide. This one serves as a passivation against further oxidation. This is the reason why such heating elements can not withstand well operation in reducing atmospheres. This oxide makes electrical contact difficult, hindering purely mechanical attempts to rejoin a broken wire. Such joint tends to be mechanically weak and electrically failure-prone. The repeated heating-cooling cycles also serve to loosen formerly tightly twisted wires.
Welding of resistive wires is difficult. Spot welder did not help in making a satisfying joint with high mechanical integrity. Soldering is out of question due to the high temperatures the heaters operate at, vastly exceeding the melting point of the solder.
The remaining easy way of metal joining is brazing, a method similar to soldering but operating with alloys with higher melting point.
Borax was used as a flux, for its low cost, good availability, and ability to shield the surface from oxygen and to dissolve metal oxides, and thus promoting wetting with the molten metal.
Silver was unsuccessfully tried as the brazing metal. While its melting point of above 900 °C could provide reasonable performance in lower-temperature applications, it turned out that it steadfastly refuses to wet the wires. No wonder, when iron and nickel are involved; silver does not wet these.
A successful brazing material was brass, an alloy of copper and zinc. Copper has excellent wetting properties towards iron and nickel, therefore, when melted, it readily flows over the resistance wire. The melting point of about 900 °C is also favorable for resistance to decent temperatures encountered in lower-temperature heaters. Brass wire in a range of thicknesses can be obtained in art supplies shops catering to beadworking audience.
Method
The broken wire is somewhat straightened to ensure good access to both ends of the wire. The ends are twisted together. Then they are heated with a torch and dipped into powdered borax; some crystals adhere to the hot wires. Heat again, repeat until the joint is satisfactorily coated. Heat yet more; the crystals of borax will puff up as they lose crystal water, then they will melt into a glass-like substance, forming a bead-like droplet over the twisted wires.
The brass is added to the joint in the form of a wire. Either it can be fed to the red-hot joint in the form of a wire, or the brass wire can be twisted around the joint before it is coated with borax. Each approach is good for different setting, try both, find what suits you better.
The joint is thoroughly heated to orange-hot glow. The brass should melt readily, forming a shiny drop under the molten borax, then soaking between the twisted wires. This moment indicates a successful joint, well-wetted with brass. The brass should not form discrete droplets or beads, nor the wetting of the wires should be spotty and uneven. Good wetting makes a relatively thin layer of brass spreading out over the underlying metal.
Then the joint is allowed to cool, solidified borax is carefully removed (it behaves like glass, it is hard and fragile; it can crack suddenly and the dissipated mechanical energy could break the wire again, ruining the effort), excessive length of the joint is cut off, and the wire is folded back to fit the heating element again.
