With the exception of the Type T copper leg, thermocouple alloys do not solder easily. It is possible to solder these materials, but very aggressive flux is typically needed to remove oxides from the surfaces during soldering.
The bead at the end is just fine, but the wires themselves have been damaged. A search match on Google says that thermocouple materials may not be solderable, that a metallurgical bond will...

The first experiment compared the accuracy of thermocouple attachment methods by securing multiple thermo-couples to a large, metallized test pad on a bare test PCB (Figures 1 & 2).

Moving forward, it's essential to keep these visual contexts in mind when discussing Thermocouple Fine Soldering.
Remove the soldering iron and apply a dab of soldering paste. The paste will instantly melt and wick into the sensor wire/solder-pad interface, quickly hardening and bonding the two together.

Yes, you can solder the junction, and it's better than twisting the wires. Even if it becomes liquid it will still work, but it would be good to stay below that temperature.