Unplug the lamp from the wall.
Pry the socket from the base using the end of a flat screwdriver.
Loosen the set screw at the socket base with a screwdriver.
Pull the wires out of the socket base.
Set the new socket base onto the lamp. Pull the wires through using a pair of pliers.
Strip 1/2 inch of insulation off the wires using a wire stripper.
Tie a knot in the two wires coming out of the lamp so the knot sits firmly in the new base.
Connect the neutral wire, which is usually indented or has a color strip, to the silver screw on the pull chain socket. Connect the other wire to the other screw.
Reassemble the lamp, plug it in and test.