It turns out the Internet Explorer doesn’t properly support CSS’s position: fixed. Google led me to the following:
- How To Create – Making Internet Explorer use position: fixed;
 - doxdesk.com: software: fixed.js
 - Fixed Positioning for Windows Internet Explorer
 
The DoxDesk solution looks promising and simple, but I think bugs elsewhere in my layout are preventing it from working. It’s time to start again from scratch.