Hello Martin - I have the Apple KB and have also upgraded to iOS 5.0.1 - no problem for me at the moment. Please let us know whether this is happening on just a single program, e.g. Safari while on the WEB, or on multiple apps requiring typing?
The usual 'trio' of initial options, i.e. 'force close' the app in question, 'Restart' the iPad, and if the latter is not successful, then 'Reset' the device has been suggested, but if not all done, you might want to give those a try - no harm done.
Do you have other Bluetooth (BT) accessories such as headphones and/or speakers? If so, are these working fine? Finally, make sure that your batteries in the KB do not need replacement. Good luck and post back w/ further questions, as needed!