Here again are the steps to replace the cord: 

1. Buy some new cord.
2. Open up the housing which holds the spring-loaded starter pulley. 
3. Remove the starter pulley (if necessary)
4. Remove the old cord 
5. Attach new cord to pulley, probably with a knot. 
6. Wind new cord two or three times around the pulley. 
7. Thread the cord through the hole in the metal pulley cover. 
8. Re-attach the pulley to the engine motor or clutch 
9. Fasten the metal cover. 
10. Give the new cord a few pulls to test the recoil. 
11. Reattach the little plastic "T" pull handle. 
12. Mow.

Detailed instructions (continued):

On some models of mower, the string pulley is attached to the inside of the metal housing. On others, it is attached to the shaft of the mower.

This mower had it on the inside of the housing. 

I cut a nine-foot length of cord. I would trim the length later.

The cord will need to be threaded into the starter pulley and fastened with a knot. To contain the frayed edge of the cord, I wrapped it with some electrical tape. 

Next, I wrapped the cord around the pulley two times.  The pulley is spring-loaded. If you spin it with your hand, you should be able to feel recoil force building up. In an illustration on the right, I've shown the spring action as having the same effect as a stretched rubber band.

Do not wrap the cord around this pulley by spinning it. You want the recoil springs to be relaxed when the loops of string are on there. 

Not all lawnmowers will need exactly two loops of string. You are trying to get one "pull's worth" of string loaded onto the pulley.

The lawnmower in these photos used a pulley with a little channel around the outside, which was almost filled when the cord was wrapped around three times. You definitely want to avoid overfilling that channel with string, otherwise it might slip out of the channel, or rub against the outside housing.

Please continue reading page 3 of Easy, Illustrated Instructions on How to replace the starter cord on a lawnmower.