Q: How do I reinstall airplay

That MacBook is too old to support AirPlay, according to this:

Requirements for using AirPlay

If you go to Apple menu > System Preferences, click Displays, and don't see the option to "Show mirroring options in the menu bar when available," your Mac doesn't meet these requirements.

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
• For mirroring your display or extending your desktop, any of the following Mac computers with OS X Mountain Lion v10.8 or later:
  
  • iMac (Mid 2011 or later)
  • Mac mini (Mid 2011 or later)
  • MacBook (Early 2015 or later)
  • MacBook Air (Mid 2011 or later)
  • MacBook Pro (Early 2011 or later)
  • Mac Pro (Late 2013 or later)

Minimum requirements for peer-to-peer AirPlay, which lets visitors quickly and easily connect their Mac or iOS device to your Apple TV without gaining passwords and connecting to the Wi-Fi network:

• Apple TV (3rd generation rev A, model A1469 or later) with the latest Apple TV software
• A Mac (2012 or later) with OS X Yosemite v10.10 or later

from:

Use AirPlay to display video from your Mac on an HDTV - Apple Support

.

Agree with Grant. I have a mid-2009 MacBook Pro 15" and there is no AirPlay support.