In this circuit, current flowing through the coil provides a magnetic field that actuates a reed switch. The coil is made of twelve turns of 20SWG enamelled copper wire to carry the current (refer Fig. 3). The number of turns will change with the wattage of the kettle and the type of reed switch.
When the kettle is switched on, current through the coil creates a magnetic field, the reed contact closes and C2 charges through R1. Although the reed switch periodically opens at AC mains frequency, the time constant of R1 plus R2 and C1 is such that there is no voltage across C1. When the kettle switches off, the reed switch is permanently open and capacitor C1 starts charging through resistor R2. This capacitor C1 voltage makes transistor T1 conduct and the buzzer sounds (as it gets the supply path) to indicate that water has boiled. Fig. 4 shows a piezobuzzer.
Assemble the circuit on a common PCB and enclose in a plastic case. Connect mains input power supply to the circuit and 230V AC output to the kettle. Fix the unit near the kettle. Connect reed switch and its coil as shown in the circuit.