Locate the hole on the side of the door where the catch protrudes from. Use a ruler to measure the distance from the top of the hole to the bottom of the hole. Divide that number in half to find the center point of the hole. Use the ruler and a pencil to mark the center point on both sides of the hole. These points will be used later to align the larger hole for the door handle.
Draw a straight line across a piece of scrap plywood.
Measure the distance from the center of the existing hole to the edge of the door. This is how far away from the edge of the door that your hole needs to be aligned. Measure along the line that you drew on the plywood to the distance that you measured, and mark the board with a dot.
Place the center of the 2 1/8-inch boring bit on the dot and drill out the hole all the way through the board.
Secure the board to the face of the door using a clamp so that the new hole in the plywood is centered over the existing hole. Use the line you marked on the edge of the door as a guide to center the board. Secure another piece of scrap plywood to the back of the door using another clamp so that the board completely covers the existing hole.
Drill through the door using the pre-cut hole in the plywood as a guide. Stop drilling as soon as the pilot bit penetrates the back board. Place the pilot bit in the hole on the back board and finish drilling out the hole from that side.