Ala Billy Mays... add the top on to the mixer, and I used a AA rechargeable battery to power the motor. A little electric tape and put the corresponding wires on each end of the battery. The mixer should start turning as you do this. Take the mixer and place in the freezer. I'm not sure how long a battery would power this. At first I was checking it about every 15 minutes to make sure it was still powered and still operational. After the first hour I went to half-hour check-ins. After about four hours in my freezer, the spindle slowed down significantly. I'm not sure if this was a power issue, or because the ice cream was much thicker. I thought it looked ready from the outside, and tore everything apart. In retrospect, it needed to be left in a while longer, maybe with a fresh battery.
See it in action!
Nice what you did with gears, lower gear ratio should provide enough torque to stir the ice cream when it is almost hard.