If you know anything about electronics you could probably fix it. Those kinds of devices have a fusable link in series with the heater and when they exceed their rated temperature then open up to keep things from starting a fire. Typically you can purchase them at a place like Radio Shack if you can determine the temperature and current rating. Good luck.
I recommend to get a better hair straightener, I bought mine at safrons it's the Karmin G3 Salon Pro with ceramic tourmaline plates to avoid hair damages, it heats up really quick and leave my hair soft, shiny and super straight, hope this helps.