Even if servo is unmodified, I don't see how you solve the "orientation" problem without, say, crown-gears or something to translate the motion if the servo is to fit in the confines of the black box.
Ignoring the broken segment gear tooth, as a complete aside I was going to suggest a really ambitious DIY'er could use an Arduino to convert the mechanism to a linear-solenoid. That is, you "instrument" the output shaft with a potentiometer (just like a servo does). An Arduino analog-input pin reads the angle and the Arduino calculates the correct PWM to drive the solenoid to the desired angle. Of course this would burn much more DC power than a gearmotor but it would be a neat project!
As for the broken tooth, it seems the segment gear is reasonably thinner than the depth of the gear on the output shaft. Is it possible to glue on a thin piece of metal like this? I'm assuming the segment gear is unavailable or uneconomical to replace.
