what quick launch bar is a bar which represents a folder.
the buttons represent the contents of the folder, which by default are shortcuts to some commonly used programs. if you were to put a "real file" such as a plain text document, that document would appear in the bar as well.
what you need to do is create a new shortcut of internet explorer in that folder which can be done by dragging the original EXE or if you have it on your desktop, that icon, to the folder. OR you can drag it to the quick launch bar itself.
by dragging the icon off the bar, you pretty much removed the icon from the folder, thus from the bar.
it is a simple fix =)