How do I get the Social Media Plugin to show up in my Twenty Eleven child theme’s header?

You can throw a widget anywhere into your template (if you do not want to use sidebar drag + drop) by calling the_widget.

In your case you would need to put this in the appropriate spot (maybe header.php or menu.php), you will have to figure out where you want it.

the_widget('Social_Widget ', $instance, $args);

Please view the ref page on what can be used for $instance and $args (for example this is where you put the widget title, parameters, etc): http://codex.wordpress.org/Function_Reference/the_widget

You should just use a register a new sidebar though , on account of this plugin have tons of options.