Unfortunately, the Kile package in Fedora 16 appears to be broken. When trying to start the application, a dialog box pops up with the message: "No editor component found. Please check your KDE installation". The application still manages to load, but is missing all icons. Attempting to open a file causes Kile to crash.
Through trial-and-error and a nudge in the right direction from the Arch forum (https://bbs.archlinux.org/viewtopic.php?id=124428), I managed to fix Kile by installing two more packages.
sudo yum install kate-libs libkate