After downloading IOS 7 on my iPad, there is a 15-20 second delay between touching a keyboard character and the letter appearing on the screen.
Anytime iOS is seeming sluggish to me, I like to place it in Airplane mode to see if it's a network issue slowing things down. For a delay that long, I would then power down the device and restart it while still in Airplane mode.
At that point you will know if your OS or device has issues to its just doing a lot of work.
From there, I would start at http://support.apple.com/iPad and start the troubleshooting assistant to narrow done the cause.
I updated to iOS 7 today, was going off my head with the unresponsive keyboard. Big delays, looked up potential bugs on google. Go into iCloud settings and switch off the toggle switch in documents and data. You will find that this will allow you to type things without wanting to launch your ipad across the room. If you access iCloud that will enable the documents and data again. Hope this helps :0)
I just updated to iOS 7.1 and as soon as I did on my iPhone 5 and iPad Air I instantly got a 7-15 second lag anywhere the OS used a keyboard including the Lock Screen.
After a couple of days of trying all the other options the only thing that works so far is to toggle OFF contacts on any gmail account in Settings -> Mail, Contacts, Calendars. This fixes the issue right away. If I toggle contacts ON this issue comes right back.
Apple needs to fix this issue right away, this worked fine before the 7.1 upgrade.