How to automate an electric kettle?
You could use lots of options to control it from Mbed boards to the Uno with a wifi shield, to a raspberry pi to .... the list goes on.  But if it were me I would use the something like the ESP8266 which cost less than $10 usd and can be programmed with the arduino IDE (I'm assuming that is useful given the community this was posted in).  There are also lots of tutorials available online for this kind of thing.

To trigger the button you have two options: a non-destructive one and a destructive one.  For the non destructive one, use something like a solenoid in a custom mount or a servo and small lever arm to push the button.  The benefit to this is that if you change your mind you just have to move the device and you still have a perfect unharmed kettle.  Plus, you can still use it if your device breaks or malfunctions.  

For the destructive option, open up the kettle to get at the buttons or button pads and solder a wire to each side.  If you're lucky you can do this so that it can still be put back together and the original button still work.  One option is to use a relay to connect them.  You can use a small solid state relay or a larger coil relay; both would be under a couple of dollars and not large enough to be too obtrusive.  Using a relay is nice since you don't have to worry about which voltages are being used on which side of the button and it isolates the microcontroller from the kettle better.  (Actually some relays are polarized so if it doesn't work, switch the wires and try again).  Another option is to use an  optoisolator like this one.  It is polarized so if it doesn't work switch the wires.  Another option is to use a mosfet.  I believe it will work if you get an N channel mosfet with a Vgs (gate to source voltage) less than 3.3V.  Hook up the Source pin to the more negative of the two button leads and the Drain to the other.  Hook up the gate to a pin on the micorcontroller.  Also, you should put a resister between the gate and the source pins.