I just downloaded a simple, free app called Loop Video. You tap, add the video you want to loop, click "Use" to make the menu stuff go away, and voila.
The only trick is that the video needs to be in your Photos library on your device. There are tricks to get it there, but I found the simplest was to open the iPad settings in iTunes, click on Photos in the left menu and make sure the "Include videos" box is checked under "Sync Photos." I put the video in its own folder in my Pictures folder and made sure that folder was checked to be included in the sync (since I sync only selected folders). Hope this helps someone.