I know I'm a noob here, so I hope I'm not jumping into something I'm not quite understanding, but I'm coming from modifying Power Wheels (adding lights, etc)...
Why not just slap a battery on the kart and run the lights directly off that? Seems way over-complicated to try and power them off the motor, especially if you are going with LED lights. LEDs will last for hours at a time on a 12v, 12ah battery, which you can pick up for $30 or so off Fleabay. Sure, you'd need to charge it once in a while, but unless you were going to be riding all night, it would provide more than enough juice to power you for a while without all the hassle of rigging up the motor.