So the cause of your problem is, that you somehow deleted Apple Mail. You maynot want to use it; but it is needed by the OS. You'll likely have to backup your iPad onto a PC or Mac (to a local full backup), then wipe the iPad and do a full reinstall of the OS. Once that is all done, restore your backup to the iPad and you should be good.
That said, try before that to go into the App Store, search of Apple Mail and see if it is available to download. If it is, that will get you around re-imaging the iPad. But in future, don't delete default applications, move them out of the way and setup your preferred application in its stead. The issue is that default apps have often OS calls being made to/through them that then interface with your preferred app. Think of it as, a puzzle with a piece missing if you don't have the default app, you can't complete the action of making the puzzle.