I know you mentioned the upgrade to iOS 6, but I'll toss this out anyway. My problem was not related to an upgrade.
A few months ago the start button on my iPad 2 stop working unless I really mashed it. I took it to an Apple store where they tried to clean by blowing some air in it, stating before hand that they didn't think it would work. It was finally determined that the start button was physically broken.
They replaced the iPad under warranty.