Update 10/6/14: Yesterday the Fleksy keyboard app released an update that contained a Dvorak layout that can be used system wide. Over the last month there have also been a few other attempts to do this, none of which have been very successful yet.
Note: This post was written before the release of iOS 8. In iOS 8 Apple allows developers to create 3rd party keyboards that work system wide. Because of this it is likely that someone will soon create a keyboard with a Dvorak layout. If you find one please let me know.
The simple answer:
The simple answer is that you can't, but there is more. There is no option to turn the iPad or iPhone touch screen keyboard to the Dvorak layout. If there was it would be under Settings > General > Keyboard > Keyboards > Add New Keyboard > Dvorak. Unfortunately there is no option to choos Dvorak in that list. It skips straight from Dutch to Estonian. Of course it could be under English (Dvorak) since it is an English layout, but no, it is not there either (English keyboards are at the top of the list). There are three options for English layouts besides the default American (UK, Austraila, and Canada) all of which are QWERTY layouts. So don't ask me why they need three of them that are exactly the same, and they can't have one Dvorak.
A different spot:
But, then we find out that is not actually where it should be at all, because if instead of clicking on "Add New Keyboard" we click on "English" we find the keyboard layout menu. This is really where the Dvorak option should be since it is an English keyboard layout. You see two menus, Software keyboard layout and hardware keyboard layout. Under software you can see that you have three options, none of which are Dvorak.
Finally after much searching we find that Dvorak is listed under hardware keyboard layout amidst a strange and non-alphabetical list. Turning on this option will make any hardware keyboards connected to the iPad Dvorak layout, regardless of what the letters on the keyboard actually say.
Your only option:
So there you have it the only option to use the Dvorak layout is to use and external keyboard. This is what I do when I am typing a lot, like for this blog post for example. But when I don't have a keyboard with me I am stuck just using QWERTY.
What is ridiculous about the whole thing is that there is no reason that Apple could not have made it an option to have Dvorak be the on screen keyboard. Clearly iOS knows the layout because it can have the hardware option. It would not have hurt anything to make it an option. I mean there are 9 Chinese options so I don't think it would have hurt them to put in another English one.
For now use an external keyboard if you do a lot of typing, and hope that iOS 8 brings an on screen Dvorak. And remember, live better.