A thermocouple welder can be made using capacitor discharge energy. I made one for fine gauge thermocouples using a 12 V DC (variable is good) power supply connected to a 3300 μF capacitor via a 1 kΩ resistor.
The TC junction should be prepared by stripping a bit, so the two wires can be tightly twisted and cut off cleanly. Connect the negative lead from the capacitor (polarity is important) to a carbon rod using a sufficiently large alligator clip. Connect the positive lead from the capacitor to the thermocouple wires (both leads); it is good if you can connect near the junction, especially for long leads, but for < 3 m it is OK to connect to the end away from where you will weld. Wear goggles or safety glasses!
Now, with the capacitor charged, touch the thermocouple twisted wire to the carbon rod and it will spark and melt a bead. Depending on wire gauge you may need to vary the voltage or the capacitance to get a good result, and it may take a few tries to get a good weld. Energy scales with V2, so a bit more voltage delivers much more energy.
Ideally, the carbon rod would be enclosed in a small plastic tube flushed with Argon gas and the thermocouple tip would be inserted through a small hole in the side to touch the carbon rod and produce an oxygen-free weld.