The recoil starter can usually be unbolted from the engine for service. If you can retrieve the end of the broken rope, and you know how long the new rope should be, you can tape or stitch the new rope to the end of the old rope, then pull on the knotted end of the old rope to pull it out of the starter and draw the new one in. If this won't work (the old rope may be too rotted to hold together, or the pull force to wind the rope around in the starter is too much), you'll have to take the starter apart. This is nasty because you have to get the spring properly engaged with the frame and the starter plate when you put it back together, and there may be an element of personal hazard if there is tension on the spring. You may want to have the starter plate on the bottom when you lift off the cover, hopefully this will prevent everything from spilling out before you get a chance to see how it goes together. Wear safety glasses and leather gloves. Wearing a padded jacket or heavy-duty shop apron may also be a good idea.
Check whether the handle on the new rope requires the rope to be fed through its opening in the starter, and be sure it goes it is wound in the same direction as the original rope. Also, you want just enough tension on the spring to hold the rope handle up against the starter without flopping around, but not so much that the spring winds up too tight before you get to the stop with the rope pulled all the way out.
You may wish to take the starter unit to a small engine shop and pay them to wind on a new rope. :)
These are the considerations I've observed when looking over recoil starters - someone may have more detailed information on your model.
Finally, evaluate the storage conditions for the generator. A good rope should last a long time; perhaps excessive humidity, chemical fumes or exposure to UV radiation rotted out the rope. Did you put oil in the cylinder per the long-term storage instructions (Page 14 of the manual)? If not, that could result in excessive pull force. You may be able to free up the generator by removing the spark plug, putting a small amount of lightweight oil, turning over the generator using the starter spline shaft (be careful not to damage the shaft with tools, and observe which direction it's supposed to rotate when the recoil starter lets go), Change the oil if you have to use something lighter than recommended to get sticky parts loosened.
Here's a link for the manual (Page 14 is actually page 17 in the PDF viewer):