This plugin provides the same shortcodes that are found in Jetpack, but without Jetpack's activation overhead.
Your hosting provider may have trouble supporting Jetpack. For example the notorious SSL timeout when trying to connect to wordpress.com/jetpack may be getting in your way. Use this plugin to get your shortcodes back.
You may have mixed closed and unclosed [audio] shortcodes in one post. This can make WordPress skip a lot of your content and not display it.
A closed shortcode looks like this: [audio mp3="url"][/audio]
A unclosed shortcode looks like this: [audio mp3="url"]
It's relatively easy to mix closed and unclosed shortcodes when you have old pre-3.6 content you're updating. The WordPress core media manager for version 2.6 and beyond has a way to insert an embedded audio player, and it always inserts a closed shortcode. It's likely that your old content has unclosed shortcodes.
For more information see this explanation in the WordPress Codex.