The DOM Inspector is a built-in Firefox extension that "can be used to inspect and edit the live DOM of any web document or XUL application." I had never found the DOM Inspector useful by itself, but some components of the (very useful) Web Developer extension rely on it.
In an attempt to troubleshoot a performance issue, I uninstalled the DOM Inspector extension in Firefox to see if it was the culprit (it wasn't). It did not occur to me that it would be so difficult to reinstall. DOM Inspector does not appear in the list of normal extensions on the Mozilla download site. It took some looking around, but I found a good place to download the .xpi for DOM Inspector to manually reinstall: http://releases.mozilla.org/pub/mozilla.org/firefox/releases/18.104.22.168/win32/xpi/ Replace "22.214.171.124" with your version of Firefox (found under Help/About). Alternately, browse to http://releases.mozilla.org/pub/mozilla.org/firefox/releases/ then click on your version, then descend into the /win32/xpi/ directory to find the DOM Inspector install file called adt.xpi. Right-click the adt.xpi file and save it to your desktop. Once it downloads, drag it into a firefox window and click the "Install" button. Your DOM Inspector is re-installed!