I'm having the same problem when trying to mirror an iPad or iPhone to the Apple TV. I'm an IT guy and have access to every version of the iPad and iPhone so here's what I did.
Verified my Apple TV had the latest update, 6.2 I believe and had internet access.
I have an iPhone 4, iPhone 4s, iPhone 5, iPad 2 and iPad 3 in the same room as the Apple TV 3rd generation.
The iPhone 4 see's the airplay icon but nothing displays when it connects. The iPhone 5 and iPad 4 does not see the icons. The iPad 2 is the only device that can connect and mirror itself to the Apple TV. The OS version for that iPad is 6.1.2.
By the way, all these devices worked with this Apple TV but quit working in the middle of a big meeting. Thanks Apple, for making us look stupid. So it's kind of an off and on thing. There's no guarantee that this Apple TV will work. I tried using the "Remote" app from apple which requires turning on Home shareing to work. That didn't help either. I can understand if all the devices I have don't work, but the iPad 2 is the only one that does. Just doesn't make sense. Any suggestions would be greatly appreciated.