Our water kettle stopped working.
Usually a thermal fuse is used to prevent this. However, when I disassembled the kettle, there was no fuse in sight. It seems the circuit relies only on the thermometer to cut off the heater!
The problem in our case is that the thermometer is not calibrated. Water boils at what is detected as 95°C instead of 100°C. Thus if you leave the kettle unattended, it will keep boiling, never reaching 100°C, and thus never shutting off. Over time, I suspect this is what causes the wires to burn out.
To get the kettle up and running again, I cut off the burnt wire, and replaced the terminals. I also sanded the contacts. A more thorough fix would add a thermal fuse for safety.
This repair was done back in April 2019. I finally got around to creating this write up.