Ah, I'm following you a bit closer, but it should actually show savings if there are any to be seen.
If it was unable to determine the mimetype, it ought to give you a warning on the Media Library saying something like Unknown file type or Unsupported file type. The mime_content_type function is actually a fall-back and isn't really supported anymore by PHP (it still exists in most installs though). Do you see No Savings on EVERY jpg you upload? Can you give me a link to an un-optimized jpg so I can test it for myself?
Otherwise, you can use the test jpgs in this file from the JPG developers: http://www.infai.org/jpeg/files?get=jpegsr09.zip
The last suspicion I have is that perhaps your server somehow prevents output redirection which is necessary for jpegtran to function. It differs from processing of PNG and GIF images in that regard, so I'd be curious to see if your server will allow you to optimize any PNG or GIF images. You could download the logo from the gifsicle site - http://www.lcdf.org/gifsicle/logo.gif
Ironically, the logo isn't optimized, so you should see something like a 3.2 KB reduction on it.