You can replace the temp switch, pics of it can help to source a new one, although just buying a new kettle would be less hassle.
Sometimes you get poor heat transfer to the temp switch - is it tight against the element base? No corrosion?
What I've also seen is a pressure port running from the lid down into the base switch. When the water boils (and the lid is closed) the steam pressure pushes that switch for auto shutoff. It's common for it to get plugged up with lime deposits. Look closely for a hole near the top and make sure it's not clogged.
A thermal fuse is there to cover when the kettle badly overheats.