How do you put the pull cord back on a feather lite weed eater?

Hi,
You will have to separate the fan housing/shroud (WeedEater's term from their parts manual) and remove the pulley. It will require a 5/32" Allen wrench to remove the four screws. There will be another shorter screw holding a metal plate shaped like an elongated "D" at one of the bottom corners that holds the pulley in. There will be a small screw with a washer attached in the pulley that will probably have the small remaining piece of the original rope. Remove it and the and the small piece of rope. The part number for it is 530069232, but going to a power equipment dealer and requesting 3 ft. of high quality rope of size 3.5 will suffice. High quality rope will have a braided exterior and three inner core strands. With the new rope insert it; if you bought bulk, make sure the ends are burned to prevent unravelling. There will be a cuved groove that the rope fits in and is bound by the small screw/washer. Do NOT overtighten and strip. Wind the rope in a counter-clockwise direction on pulley as referenced by the side that you would see if mounted in housing. Put through eyelet in housing, into starter grip and secure with a knot. Slip the pulley into replace after winding up slack and pretension. To pretension You must wind the pulley two turns counter-clockwise to remove the slack and make the rope fully rewind when used. There could be a single notch in the pulley to facilltate this , or a number of scallops around the pulley. Holding the pulley loosely in place with a finger, take a pair of needle nose pliers and ease the rope toward you in a loop long enough to make the revolutions. Care must be taken to keep the pulley seated on the spring during this to achieve the pretensioning. Once done replace the screw and plate to hold the pulley in place. Seat the housing to the engine, a small, short pull of the rope may be necessary to get the starter pawls into position. Holding It in place, test it by pulling it slowly through to see if all works correctly. Then and only then do you reinstall the four screws that hold it together.
That does it and, if all other things are functioning properly, the machine should be startable. A note: there is no reason in the world to pull the rope to full extension, ie all the way out until it stops; that breaks starters! You only need 12 to 18" of the 3 ft. of rope to start a working machine.
Peace,
Clarke
PS. I could have repaired three of these machines in the time it took me to write this answer.