“Add Media” broken in WordPress 3.5

A little while back I upgraded to WordPress 3.5.X like a good little server admin. Sometime shortly thereafter I noticed the “Add Media” button stopped working, or it would work randomly for one post and not another. Very strange and very annoying since that is the one button I actually need (I write in HTML format, so I don’t need the rest of the WYSIWYG interface to work). After a...