Rewire the Socket
For each socket, cut a piece of lamp cord the same length as the old socket wires, then part and strip both ends of the socket wires, as well as the upper end of a new lamp cord.
Connect the neutral wire -- the one with the ridge in its insulation -- to the silver socket terminal and the other wire to the brass socket terminal. Place the socket in the lamp, feeding the wire through the socket cap.
Reinstall the socket's insulating sleeve and shell, pushing on the shell until it snaps in place.