The instructions already on the web say you have to take the curtain rod down and feed the cord through the pulley wheels, behind the sliders, and up through small holes using a bent paper clip or crochet hook. It is very complicated and daunting. The following way is easy and quick. Basically, you sew the ends of the old cord to the ends of the new cord, and use the old curtain cord to pull the new cord through the pulleys and holes, without taking the curtain rod off the wall.
Get a new curtain cord the same exact diameter as the old cord.
Measure how much you need: height plus width, times two. (Do not forget to double the sum of the height of the cord pull and width of the curtain.) Add an extra foot of cord to be safe.
At the break in the old cord, sew one end of the new cord to one end of the old cord. Sew the other end of the new cord to the other end of the old cord at the break. Use a strong, thin thread. You can wrap the thread around the break to keep the frayed ends in place. Sew through the ends of the cords at least one inch from the ends to create a strong connection, but do not add any extra thickness around the break. This junction will have to travel smoothly between the pulley wheels. If the cord is white, it will be useful to use black thread, so you can later see where you have connected the ends.
Go to the carrier(s) on the rod. If there is one curtain panel, there will be one cord carrier with two holes in the back with knotted cords. If there are two curtain panels, there will be two cord carriers, each with one hole and a knotted cord through it. Find behind each carrier where the old cord is pulled through a hole in the metal and secured with a double knot in the back. Pull carefully at one of the knots until you have pulled all the old cord out through the hole, and you have about six inches of new cord pulled through. Find the other old knot and carefully pull the old cord out through the hole until you have some of the new cord pulled through.
You can now cut off the old cord. Pull the two ends of the new cord so the new curtain cord is taut in the closed position. Tie a new double knot behind the carrier for each cord end.