I encountered a strange bug on my iPhone that has an even more bizarre solution. Here are the details in case the same thing happens to you. I have an Apple TV connected to my television and I wanted to mirror what was on my iPhone to the TV. Normally, this is very simple to do. As explained in this Apple support article, on an iPhone 4S or later, you can double-click the Home button (which displays your recently used apps) and then swipe twice from left to right to display the AirPlay icon on the bottom right of your screen. You then tap that icon, select an Apple TV that is on the same network, and turn on Mirroring. The procedure is fast and simple, and it makes everything on your iPhone screen also show up on the TV.
At least, it is supposed to be that simple. When I tried to do so the other night, I didn't see an AirPlay icon:
After confirming that my Apple TV was working and that my Apple TV and iPhone were on the same local WiFi network, I wasn't sure to do next. After a lot of searching the Internet, I finally found this thread on the Support Communities area of Apple's website that offered a bizarre solution that actually worked for me.
It turns out that you can fix this problem by going to your Settings app and selecting General -> International -> Language. Then change your language from whatever you are using now (such as English) to another language (I chose French). Then switch it back to your original language.
That's it. Now when you double-click and the Home button and then swipe twice from left to right, you should see the AirPlay icon. As you can see, it worked for me:
I am running what is now the latest version of the iOS for iPhone, version 6.1.4. I suspect that Apple will fix this in the future — indeed, iOS 7 handles the buttons for things like AirPlay in a whole new way — but for now, there seems to be some connection between changing your language and mirroring to an Apple TV.