How to make the WP 3.6 media player responsive?

Hi, I want to auto-play a local mp4 video file. So with WP 3.6?s new built-in media player, I thought that would be the best/easiest option. I’ve inserted it in the home page using shortcode. When I use width & height settings, it looks fine, as long as the screen is wide enough: http://macuclear.fiddlersites.com/client-preview - but it doesn’t resize to fit if I drag the window narrower. The theme is very responsive, but the video stays the same size/width.

I thought maybe if I just left the dimensions out of the shortcode that would allow it to adapt to the theme's CSS correctly. But without dimensions it only displays the video small and the player goes outside the area it should be inside, on the right: https://www.evernote.com/shard/s26/sh/37b28a4b-1c72-4a4f-8d7a-00cd328fe5e5/1d868cc56d6369a16b9216842a31e16a

Do you have any ideas about how to make it responsive?