My problem was similar for the Cuisinart KUA-17. I clicked down the on switch and it stayed down. However, the light and power together seemed finicky. That is, the light came on if I positioned the kettle in the base correctly, but then it sometimes went off without triggering the switch to pop up. When the light went off while the switch remained down the water never boiled, but when I was lucky to position it such that the light remained on, the water did boil and the switch popped up ... as it should have. It seemed to be getting more and more finicky and difficult to position correctly ... so I took it apart.
First of all, the base has screws that take a triangle screwdriver which I don't have. So I found a small flat-headed screwdriver (like one you'd use for glasses) that just fit along one side of the triangle. I was able to get three base screws out this way and since I expected to need to do this again in the future, I used my dremel to grind a larger flat-headed screwdriver slot in each. It's been about a year and a half since I fixed it last time and this time I was glad I had the slots.
Once you get the plastic base off, you'll see the mechanics of the switch. There are two wires that both turn on the light and power the heater element. Where the wires connect to the switch mechanism, there are two metal tabs that stick out in opposite directions from the center. When the switch is pushed down, you can see these two tabs move. They are each moving to touch a contact and complete the electrical circuit. My problem was they were not making a good contact. The contacts are hidden behind the tabs. All I had to do was bend the contacts one millimeter closer (i.e. as little as possible) so that they would make a good contact. The only other trick was I had to remove the switch mechanism from the bottom of the kettle to do this. Then I just used the same small flat-headed screwdriver to bend the contact as in the picture below.