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Here are my standard directions for replacing the rope on your recoil. You need to make sure that you are prewinding the spring before installing the new rope.
Remove the four bolts that hold the recoil housing onto the engine. This is as far as you need to disassemble it. Turn it over and you should see the knot from the old rope get some needle nose pliers if necessary and pull the old rope out by the knot. This is by far the hardest part. Once all the old rope is gone begin to prewind your pulley by turning it in the same direction as if you were pulling out the rope to start it. Wind it up as far as you can until it stops. Now turn it back until the hole in the pulley and the hole in the housing line up. Insert your new rope through the housing hole and straight through he hole in the pulley and tie a knot. Holding onto the rope, allow the spring tension to pull the rope into the housing. Reinstall on your engine.
One of two things has happened. Either the spring inside has broken or it has come undone from the fingers that hold it together. At each end of the spring there is bend. One end hooks around a tab on the recoil housing, the other end hooks around a tab on the recoil pulley. A lot of these recoils do not come apart, but some do. Some have a screw in the center that you can remove and some have a squeeze tab in the center of them that allows them to come apart.