How do I install a battery kill switch in my 2010 Corolla S?

What do you want to kill, the starter or all battery power?

Then you can't do that from the ignition switch. It has to be done from the battery cable and it would be quite obvious if you're trying to hide it from someone.

Your car already has a security immobilizer.

I mean it already has a factory security system that disables the engine.

You don't have to activate anything.
THEFT DETERRENT / KEYLESS ENTRY: ENGINE IMMOBILISER SYSTEM (w/ Smart Key System): SYSTEM DESCRIPTION

ENGINE IMMOBILISER SYSTEM DESCRIPTION

(a) The immobiliser system is a theft deterrent system that determines whether to disable starting of the SFI system based on a comparison of the key ID codes and the vehicle pre-registered code. The immobiliser system compares the vehicle certification ECU (smart key ECU assembly) pre-registered ID code with the ID of the key-embedded transponder chip. If the ID codes do not match, the immobiliser system is activated and the SFI system cannot be started. The certification ECU (smart key ECU assembly) manages communication with the ECM, main body ECU (instrument panel junction block), steering lock ECU and ID code box (immobiliser code ECU). When the ID codes of the transponder chip and certification ECU (smart key ECU assembly) match, the certification ECU (smart key ECU assembly) authorizes the SFI system to start.

FUNCTION OF MAIN COMPONENTS

SYSTEM FUNCTION

(a) Using the entry function
(1) When the driver (or passenger) is sitting in the vehicle while carrying the key, and the engine switch is pressed while the brake pedal is depressed, the main body ECU (instrument panel junction block) recognizes that the engine start operation has occurred and sends a certification request signal to the certification ECU (smart key ECU assembly). Upon receipt of the certification request signal, the certification ECU (smart key ECU assembly) sends a request signal to the indoor electrical key oscillator. Upon receipt of the request signal, the indoor electrical key oscillator sends a request signal to detect if the key is inside the vehicle. When the key receives this request signal, it answers by sending an ID code containing a response code through the glass antenna to the door control receiver. Upon receipt of the ID code, the certification ECU (smart key ECU assembly) analyzes the code. If the interior certification passes, then the main body ECU (instrument panel junction block) sends a certification pass response signal. When the main body ECU (instrument panel junction block) receives this signal, the ACC relay is switched on and the IG1 and IG2 relays are switched on in sequence. At this time, the engine switch indicator illuminates in green. Then the certification ECU (smart key ECU assembly) checks that the power source mode has been changed and sends a steering lock command signal to the main body ECU (instrument panel junction block). After receiving this signal, the main body ECU (instrument panel junction block) supplies power to the steering lock actuator. Then the steering lock ECU confirms that the certification ECU (smart key ECU assembly) is certified (via the ID code box (immobiliser code ECU)) and drives the steering actuator motor until the steering is unlocked. After unlocking the steering, an unlock completed signal is sent to the certification ECU (smart key ECU assembly). Upon receipt of this signal, the certification ECU (smart key ECU assembly) sends an unset command signal to the ID code box (immobiliser code ECU). Once this signal is received, the ID code box (immobiliser code ECU) confirms that the certification ECU (smart key ECU assembly) is certified, sends an immobiliser unset command signal to the ECM, and sends a security indicator light off signal to the certification ECU (smart key ECU assembly).
(b) Not using the entry function (when the key battery is depleted)
(1) When the driver is sitting in the vehicle while carrying the key and depressing the brake pedal, the main body ECU (instrument panel junction block) recognizes that the stop light switch is on and sends a key confirmation request signal to the certification ECU (smart key ECU assembly). Upon receipt of this signal, the certification ECU (smart key ECU assembly) drives the immobiliser amplifier built in the engine switch. At this time, the engine switch sends an RF wave communication signal to the immobiliser. If the driver holds the key up to the engine switch at this time, the engine switch receives the immobiliser RF wave signal and responds by sending a radio wave signal. When the engine switch receives the radio wave signal from the key, it duplicates the signal and sends an ID code to the certification ECU (smart key ECU assembly). Upon receipt of the ID code, the code is analyzed. If the certification passes, the certification ECU sends a key certification pass response signal to the main body ECU (instrument panel junction block) and a sound buzzer request signal to the meter ECU. When the main body ECU (instrument panel junction block) receives this signal, the ACC relay is switched on and the IG1 and IG2 relays are switched on in sequence. At this time, the engine switch indicator illuminates in green. Then the certification ECU (smart key ECU assembly) checks that the power source mode has been changed and sends a steering lock command signal to the main body ECU (instrument panel junction block). After receiving this signal, the main body ECU (instrument panel junction block) supplies power to the steering lock actuator. Then the steering lock ECU confirms that the certification ECU (smart key ECU assembly) is certified (via the ID code box (immobiliser code ECU)) and drives the steering actuator motor until the steering is unlocked. After unlocking the steering, an unlock completed signal is sent to the certification ECU (smart key ECU assembly). Upon receipt of this signal, the certification ECU (smart key ECU assembly) sends an unset command signal to the ID code box (immobiliser code ECU). Once this signal is received, the ID code box (immobiliser code ECU) confirms that the certification ECU (smart key ECU assembly) is certified, sends an immobiliser unset command signal to the ECM, and sends a security indicator light off signal to the certification ECU (smart key ECU assembly).