I can't get the airplay to work on my desktop

Airplay works on the following Macs, if you don't have one of these then AirPlay will not work.

Requirements for using an AirPlay display

Minimum requirements if your Mac and Apple TV are on the same network:

• Apple TV (2nd generation or later) with the latest Apple TV software
• Any of these Mac computers with OS X Mountain Lion v10.8 or later:
  • iMac (Mid 2011 or later)
  • Mac mini (Mid 2011 or later)
  • MacBook Air (Mid 2011 or later)
  • MacBook Pro (Early 2011 or later)
  • Mac Pro (Late 2013 or later)

Sorry, I see you can see the icon. You did not indicate what occurs when you click the AirPlay icon, so please click Use AirPlay to see your Mac screen on an HDTV - Apple Support and ensure you are following the directions.