Since there was some confusion on what to update and where, I have made it as easy as possible. I installed a fresh copy of WordPress 3.5.1 and a fresh copy of the Sponsors Carousel and updated with my instructions.
You can get a copy of the files for my Sponsors Carousel here: https://github.com/davidwickman/Sponsors-Carousel/commit/77d6ada2b6777caed0d2bd1f33ff136f71ea2213
I did an initial commit for the original files and then one for my single change.
Here is a screenshot of the current back-end with the Add Media button now visible.
https://i.imgur.com/1BFcE2J.png
I will also try to update the back-end interface to also mimic a native WordPress Plugin later.
Hopefully this helps out, and original developers, feel free to incorporate any changes.