Replacing the starter rope on most mowers is a little tricky and it may be to your advantage to simply purchase a new rewind starter assembly. Not knowing your specific model number I can only provide general information. It appears that your mower has a Tecumseh engine on it.
To reinstall or replace just the pull rope you will need to remove the starter rewind.
Remove the 2 screws that hold the starter assembly to the engine.
Lift the starter off the engine.
If the old rope broke off at the pulley you may have to pull the staple out of the end of it,
Without the rope installed rewinds the starter pulley 5 to 8 times.
Place a screw driver thru the vanes of the pulley and the top of the housing to keep the pulley from moving and align the hole is the housing with the slot where the staple was pulled from.
Since the pull rope is probably nylon burn the ends that goes into the start assembly and taper it for ease of installation. You can also use a paper clip attached to the end of the rope to act a as a guide to make it easier to install.
Thread the rope into the pulley and knot the end.
Tie a slip knot in the pull rope where it enters the housing.
Carefully remove the screw driver and allow the pulley to retract the rope.
Untie the slip knot.
Reinstall the housing to the engine.
If there is an eyelet on the handlebar loosen it enough to be able to get the pull rope thru it.
The pull rope size is usually (98" X 9/64" DIA.)