Q: Airplay over ethernet from Mac mini?

About AirPlay and Airplay Mirroring

AirPlay Mirroring requires a second-generation Apple TV or later, and is supported on the following Mac models: iMac (Mid 2011 or newer), Mac mini (Mid 2011 or newer), MacBook Air (Mid 2011 or newer), and MacBook Pro (Early 2011 or newer). For non-qualifying Macs you can try using Air Parrot.

Several Apple Articles Regarding AirPlay

Apple TV (2nd and 3rd gen)- How to use AirPlay Mirroring

How to set up and configure AirPort Express for AirPlay and iTunes

About AirPlay Mirroring in OS X Mountain Lion

iTunes 10- About playing music with AirPlay

Troubleshooting AirPlay and AirPlay Mirroring

Using AirPlay

Thanks to the $15 Beamer, AirPlay streaming is still possible on Macs  that do not support Airplay and mirroring.

Another solution is the Air Parrot.

I'm afraid not. I know this is a pain, but you could try reinstalling Mountain Lion.

Reinstalling Lion/Mountain Lion Without Erasing the Drive

Boot to the Recovery HD: Restart the computer and after the chime press and hold down the COMMAND and R keys until the menu screen appears. Alternatively, restart the computer and after the chime press and hold down the OPTION key until the boot manager screen appears. Select the Recovery HD and click on the downward pointing arrow button.

Repair the Hard Drive and Permissions: Upon startup select Disk Utility from the main menu. Repair the Hard Drive and Permissions as follows.

When the recovery menu appears select Disk Utility. After DU loads select your hard drive entry (mfgr.'s ID and drive size) from the the left side list.  In the DU status area you will see an entry for the S.M.A.R.T. status of the hard drive.  If it does not say "Verified" then the hard drive is failing or failed. (SMART status is not reported on external Firewire or USB drives.) If the drive is "Verified" then select your OS X volume from the list on the left (sub-entry below the drive entry,) click on the First Aid tab, then click on the Repair Disk button. If DU reports any errors that have been fixed, then re-run Repair Disk until no errors are reported. If no errors are reported click on the Repair Permissions button. Wait until the operation completes, then quit DU and return to the main menu.

Reinstall Lion/Mountain Lion: Select Reinstall Lion/Mountain Lion and click on the Continue button.

Note: You will need an active Internet connection. I suggest using Ethernet if possible because it is three times faster than wireless.

open system prefs and click on displays. on the bottom left  check the box that says " show mirroring options in the menu bar when available". then in the upper right corner of desktop an icon should appear that looks like a rectangle with a triangle on it