If you do decide to keep the saw, just put a new piece of rope on. Once the rope starts to break, it'll happen again right away, so just replace it. Go to the hardware store and take the old cord with you, you want to get the exact same size. Bigger rope will be a problem because the same length won't fit on the recoil, smaller rope will break easily. Look for rope designed for recoils, not cloths line or something.
Four screws hold the recoil cover on, look how the old rope was on and copy. Try to get as long a piece of rope on as will fit on the starter drum. After the rope is completely installed, you need to get recoil tension. Simply grab the rope from the side of the starter drum and get about 6" of slack, just before the rope exits the cover, and spin the starter drum about 7 times around by holding the rope. If you look there is a slot on the starter drum to help the rope fit as you spin it around. This will build tension.
Don't go too many times around, just enough so the handle snaps back and just stands up.
Then reassemble.