Element does doesn't switch off.

By staying on do you mean the "hot" light stays on or do you mean the burner is still glowing red when you shut the switch off?
if it is just the hot light then the sensor on the burner has failed and will need to replace the entire burner and sensor, comes together as one piece.
if it is still glowing then you will have a stuck contact in the selector switch and that switch should be replaced.

What your going to have to do is replace the infinate control for that burner. ( thats the control you turn to sw it from high to low heat )
good luck