Run this command and tell us if it turns it off:

$ sudo su -c "echo 0 > /sys/devices/platform/sony-laptop/kbd_backlight"

You will have to type in your password to try it. To open up a terminal, just type Ctrl-Alt-T or open it from the Unity menu.

I'm just going to go ahead and assume this command works and show a nice little script that will automagically turn the keyboard backlight on and off when you unplug the battery. If the command doesn't work....well the script'll be here for future reference then.

Run the command gksudo gedit '/etc/pm/power.d/99_kbd_backlight'. Type in your password.

Then paste the following into the gedit window:

#!/bin/bash

export DISPLAY=:0.0

if on_ac_power; then

    echo 1 > /sys/devices/platform/sony-laptop/kbd_backlight

else



    echo 0 > /sys/devices/platform/sony-laptop/kbd_backlight

fi



exit 0

Then run this command:

$ sudo chmod +x /etc/pm/power.d/99_kbd_backlight

You may need to restart your computer for this to take effect. Theoretically, this should turn your keyboard backlight on and off when you plug and unplug your charger.

Edited the script from this source: http://www.techytalk.info/ubuntu-disable-enable-compiz-battery-ac-script/

EDIT: This may be a more "proper" answer:
Keyboard backlighting not working on a Vaio VPCSB11FX