I'm surprised no one else has suggested this yet since it's pretty simple, but I fixed my iOS6 issues with a reset. Admittedly, this was kind of a Hail Mary pass that even I didn't think could work in a million years, but my iPad 2 is about 98% back to normal ("normal" being its wondrous state of responsive, fast functionality in iOS5).
Make sure you have everything Cloud backed up (obviously), then perform the reset using the "Erase All Content and Settings" option. When the device reboots and brings you to the "Set Up" walk-through, make sure you set up as a new iPad, not restoring from a previous backup (remember, all your data and apps can be re-downloaded from the Cloud).
Granted, I spent about three hours after this re-downloading apps, magazines, books, and music and re-inputting my Mail settings and Dropbox configuration, etc., but it was all worth it.
The only thing I did differently this time around is that I opted not to re-download any apps I hadn't used in more than six months, so goodbye Grand Theft Auto, Goodreader, and a bunch of other space hogs. My completely unscientific, uninformed, wild-guess theory is that iOS6 requires more of the disk space for temp files during its operation (that may be a very Windows-like perspective). I have about 70% of my disk space free (pre-reset it was more like 10%) and it's purring like a kitten.