I love the ability to AirPlay to the Apple TV so I can understand the desire for it to work. If you don't see the AirPlay icon in iTunes, please check this out.
Use AirPlay to wirelessly stream content from iTunes on your computer - Apple Support
When an AirPlay enabled device is available, you'll see to the right of the volume slider in iTunes. If you don't see the icon, follow these steps:
- Check that you have the latest version of iTunes and Apple TV software.
- Connect your computer and your other device to the same Wi-Fi network. If you're using Ethernet, connect both devices to the same router or hub. If you're using both Wi-Fi and Ethernet, connect your AirPlay device to the same router that your computer uses for Wi-Fi.
- Restart your computer, AirPlay device, and network router.
- Since some Airplay devices include an on/off option for AirPlay, look for the option in Settings and make sure the feature is on. Check the device's user guide for more information.
Let us know how that goes.