Driving a drill bit with a stepper is a bad idea. Steppers are built for very precise positioning, NOT torque. The instant you try to drill into anything with a stepper, the stepper will stall.
If you have that stepper, try running it and stopping the shaft with your fingers (or a pair of pliers, if your motor is strong enough) and compare its torque to that needed to drill into wood.
Perhaps a better idea for your project would be a regular motor running that drill bit, and a stepper controlling the position of that bit (but I'm guessing here - we need to know what your project is), like this: https://en.wikipedia.org/wiki/CNC_wood_router
Ok, I see what you are doing now. A stepper isn't such a disastrous choice as I thought - it may be able to move the food, unless a piece gets stuck. With such low torque, probably just duct tape will work.
However, remember that steppers are designed for very precise positioning. A DC gearmotor, like in the video, is almost certainly a better choice. Steppers introduce added complexity.