The most likely cause is a broken recoil spring. This spring is located under the cover of the mower starting cord. Replacement are available at small engine repair shops.
It could be that the coil inside that is supposed to draw back the string is worn out or broken. After many years of pulling the coil becomes loose and can not pull the cord back inside.
The cord wraps around a spindle which is connected to a spring. Pulling the cord winds the spring, which pulls the cord back when it unwinds. Your spring is broken.
Most likely the recoil spring has been damaged where the cord goes into the engine. Also check to make sure the cord is not binding external to the engine.
