I've had this issue as well when using a Bluetooth keyboard with my iPad Air (iOS 8.0.2).
As soon as I disconnect the keyboard, the software keyboard will pop up on screen, regardless whether the context allows text entry.
The keyboard responds to input (key press), but even when having an application such as Notes open, typing doesn't do anything. I've noticed that when you split the keyboard in two, it will actually reveal a second keyboard behind it for a split second.
The only workaround I've found is to hard reset the iPad by holding both the home button and the lock button for ten seconds (until the Apple logo appears). After that, the permanent on-screen keyboard disappears.