On your iPad, iPhone or iPod touch, navigate to Settings > General > International > Language and change to a different language… The device will go through a language switch/reset process, and your AirPlay icon should be restored.
Next navigate back to Settings > General > International > Language (translated into the new language, of course) and switch back to your original language.
Change languages: love it! Fortunately, I remembered enough high school French to try it, but it seemed silly and didn’t work. But since it has worked for many others, it gets my vote for the most interesting work-around ever.