How much is your time worth? And how much do you care for this piece of fine equipment? I would just spend the 130 euros and get a new heater — you should get the upgraded SX-90-style heater, which will allow you to use the superior SX-90 tips. (Your existing SX-80 tips remain compatible.) The part number stayed the same, so any spare you get now should be the SX-90 style.
Also, make sure the heater connector color matches: I think the SX-80 always has a black connector, but make sure. Early SX-90 also had black, but later SX-90, and the SX-100, use a blue connector. (This is about the color of the internal heater connector, not the DIN connector on the cable.)
Black connector heater: 6010-0107-P1
Blue connector heater: 6010-0163-P1
Also, given the heat damage you have, I would also make sure the heater connector in the handpiece isn’t damaged.