To start a lawn mower with a broken cord or broken rope you can remove the top part from the engine where the cord or rope is.
Then wrap another rope around the round circular thing and then pull on it and the mower should start.
Another option is to put a socket on a drill to help spin the engine over and the engine should start that way as well.
You just have to be careful when doing it with the drill and remove the drill as soon as it's starts.
I've had to use a drill before to start a lawn mower when I couldn't replace the cord or rope right away and needed my lawn mower.
However the best thing to do is to replace the starter cord or starter rope which is pretty cheap to do.
You can buy replacement starter ropes for lawn mowers on ebay, online and at hardware stores pretty cheap and it's pretty easy to replace the broken starter rope on the lawn mower.
Now I make sure to replace the broken starter rope on my lawn mower and small engines when I see that they are getting worn out and about to break.
It's better to replace them before they break and you need it than to have it break and need it.