My iPad 3rd generation was powered off in the middle of an update to iOS 6.1. Now it is boot looping constantly, and it will not reboot, and is not detected in iTunes. How do I fix this?

Sounds like you need to use recovery mode now in order to restore your iPad. You will need to use iTunes for this.

• Disconnect the USB cable from the iPad, but leave the other end of the cable connected to your computer's USB port.
• Turn off iPad: Press and hold the Sleep/Wake button for a few seconds until the red slider appears, then slide the slider. Wait for iPad to turn off.
• If you cannot turn off iPad using the slider, press and hold the Sleep/Wake and Home buttons at the same time. When the iPad turns off, release the Sleep/Wake and Home buttons.
• While pressing and holding the Home button, reconnect the USB cable to iPad. When you reconnect the USB cable, iPad should power on.
• Continue holding the Home button until you see the "Connect to iTunes" screen. When this screen appears you can release the Home button.
• If necessary, open iTunes. You should see the recovery mode alert that iTunes has detected an iPad in recovery mode.
• Use iTunes to restore iPad.

Here is the site from which the directions are copied.