Okay, there are two things going on.
First, since you are running WordPress 3.6, you don’t need to run any other media player plugins. Deactivate Widgetkit and use the following shortcode to add your audio player:
The reason you aren’t seeing the play button on your players is that your web server is not serving out SVG images correctly. The MediaElement.js player uses a single SVG image named controls.svg for all the buttons on the player. Your web server is not serving the SVG image correctly. Here is where your SVG image is located on your website: controls.svg.
Click on that link, and compare to this link: controls.svg on wp.com
It could be that you don’t have the correct MIME type installed for SVG files. It should be: image/svg+xml