If your pull cord rope keeps breaking, it can be a couple things that need checked out.
First, are you using good quality pull rope each time you replace it? There are some very basic white ropes that do not have a lot of strength and will break quickly. A braided rope that is heavier duty will hold up much longer and suit your lawn mower application quite well.
Another problem that is typical is that your pull starter rope guides have developed a sharp spot or the blower cover itself has a sharp piece of plastic that is cutting the rope. Carefully check the guides for any sharp edge. Remove the blower or starter from the mower and check the starter rope guide for nicks or sharp edges both on the outside and the inside.
Smooth out any edge you find with some emery cloth or sand paper and form a nice smooth edge for the pull rope to pull through.
If your knot keeps breaking at the pull cord handle, you can either double knot it or also place a small washer in the handle and pull the rope through it. The washer serves as a support for a hole that may be enlarged by the pull rope. Tie a double knot in the rope and it should be good to go!