My Delonghi Scultura kettle wont switch off automatically when the water is boiled.
Most self stopping kettles have a bi-metal piece, generally in the handle or top part, which, when boiling, gets one side of the metal to expand more than the other, causing it to bend and mechanically turn off the switch.
There are many causes that it can stop working- iron fatigue, corrosion, jammed switch etc that can make it stop working. In general, it <strong>wouldn't</strong> be financially worthwhile to repair it, however many people still use them when they stay in the room, since they still do a decent job of boiling the water quickly. Typically they have a safety, so the thermo-fuse will shut it before a fire starts- on your own risk.
It's usually not so expensive to get a new one, so I'd go for using it temporarily til I get around to getting a new one!