How does the plugin know to find/use my css?
Jetpack doesn't actually do anything with your site's CSS: the print button only sends the "print" command to your browser, and your browser then looks for a stylesheet.
Could you try replacing #menu by .menu, since that container uses a class, and not an ID? You might also want to add #hellobar-banner, #hdr_img_container_other p to the list of elements to hide.
I hope this helps.