Remove the cable drum by sliding it off from the head plate. Inspect the cable drum and notice where the cable enters and exits. The exit is marked by a set screw that holds the cable in place.
Mark where the cable enters the cable drum -- where the cable first meets the cable drum -- by wrapping the cable with a strip of electrical tape at the point of entry.
Remove the ball bearing plate with your screwdriver. This exposes the hole the cable feeds through to reach the other side and the set screw.
Loosen the set screw on the other side of the drum and remove the cable from the drum.
Line up the new cable with the old cable. Start from the beginning of the cables, which start with a small loop and run them together through your squeezed hand (this ensures the old and new cable have the same length). Continue running through until you reach the black electrical tape. Mark the new cable with black electrical tape.
Insert the new cable into the cable drum, threading it through the opening and through the small hole located underneath the ball bearings. Feed through until the black electrical tape meets the drum.
Fasten the new cable by tightening the set screw and reinsert the ball bearing plate by pressing it in.
Remove the excess cable, located on the set screw side of the drum, with a pair of cable cutters. Leave about 1 foot of extra cable.