Apple TV LED continues to flash rapidly following update process.

Welcome to the Apple Community.

If your problem persists get yourself a micro USB cable (sold separately), you can restore your Apple TV from iTunes:

1. Remove ALL cables from Apple TV. (if you don't you will not see Apple TV in the iTunes Source list)
2. Connect the micro USB cable to the Apple TV and to your computer.
3. Reconnect the power cable (only for Apple TV 3)
4. Open iTunes.
5. Select your Apple TV in the Devices list, and then click Restore.

(You may already have a micro USB cable if you have a camera or other digital device)