Dashboard widgets screen not showing all widgets, not showing sidebars at all

This problem started a few days ago with two of my self-hosted blogs but not a third. Widgets stopped showing up in the sidebar on the right, and if I go to the dashboard and click Appearance and then Widgets, as shown in this screenshot —
http://tachyonlabs.com/miscimages/wordpress-problem.jpg
— only three of the widgets show up, the sidebar areas to drag to don’t show up at all, and if I click the “Screen Options” button nothing happens. Also, JetPack stopped collecting stats on visits to those blogs.

I had updated to the latest version of JetPack a week or so ago, but everything was fine immediately afterward, and I had not made any changes to my blogs at all since then.

I tried restoring my blogs from backups I had made just before updating JetPack — it not only did not help, but also I got the “Fatal error: Cannot redeclare class Wickett_Twitter_Widget” error which I found a lot of references to online.

In trying to research this online I found this page —
http://en.forums.wordpress.com/topic/dashboard-widgets-page-doesnt-work-anymore?replies=123
— where a whole bunch of people whose blogs were hosted on WordPress.com were experiencing something somewhat related two months ago, but it didn’t apply to self-hosted blogs.

Personally my guess is that this is probably a JetPack issue, but I wanted to post it here as well.