Maybe that's why mirroring your smartphone or tablet is only officially supported on Android - if you want to get your iPhone or iPad's display on the big screen, the best option is to buy the somewhat expensive Apple TV.
However, it works just a little bit better if you're living within Google's ecosystem - just like Apple's devices work better together. Google's smart-TV dongle, the Chromecast, is a very versatile gadget that can be used to stream video content from the Internet through a wide variety of devices and sites to a big screen TV.