Type k will work up till it melts ( and then a little more until the drop falls off), and even if the response is not linear it is at least monotonic. I have often repaired them with an acetylene torch to weld the wires back together, or to weld them back to the steel plate they were measuring. The insulation almost invariably was the limiting item, unless you used a filled magnesia tube with a steel jacket, or a sintered ceramic sleeve. Come back to one on a heater and find the first few cm are bare wire, the glass fibre having flaked off, but it still works as long as the bare wires do not touch. If they did the controller would cook the heater trying to get it to setpoint.