The Drill is one of six tools that Henry receives inside a cake at the start of Escaping the Prison, and one of three items that actually works. When chosen, Henry uses the drill to dig a hole in the floor of his cell and reach the room below. The game continues from there towards the Badass Ending.
Press the donut on the shelf to the left and it will say: DONUT+1. This is a part of the Donut Want medal.