I would guess that the control swich is defective - if it was the heating element you would probably see the "heating up" neon.
I believe that your chances of repairing it or of having it repaired are very slim and that a new iron may be the most realistic thing for you.
I have long since given up trying to repair my defective irons - anything over 2 years use is good. Modern safety legislation causes them to be designed and made to be difficult / impossible to dismantle - to avoid the risk of death and injury to users. Kettles and toasters also tend to fall into this impossible to mend category.
Peccavi
November 2009