Wrapping a block in jQuery Accordion
Easy Solution 1

As it turns out the accordion blocks does work. However if you're just using it for a single block it won't automatically because the default jQuery accordion only works with multiple objects.

Install Accordion blocks and open the accordion_init.js file. It'll look like this

(function($) {
  Drupal.behaviors.accordion_blocks = {
      attach: function() {
        $('.accordion_blocks_container').accordion({header: "h2", autoHeight: false, });
      }
  };
})(jQuery);


You need to add the property, collapsible: true, as shown below. I've also removed the animation and made it start closed by default.

(function($) {
  Drupal.behaviors.accordion_blocks = {
      attach: function() {
        $('.accordion_blocks_container').accordion({header: "h2", autoHeight: false, collapsible: true, active: false, animated: false});
      }
  };
})(jQuery);


Possible Other Avenue

I also managed to get jQuery accordions working, it has to have a certain HTML structure (see the UI documentation) which is difficult to implement around blocks thus why I took the easy route. 

But if you fancy trying it the following code will enable jQuery accordions on your website for your theme at a basic level. This code goes into the template file.

function themename_preprocess_page (&amp;$vars) {
    drupal_add_library('system', 'ui.accordion');
    drupal_add_js('jQuery(document).ready(function(){jQuery("#accordion").accordion({ collapsible: true, active: false, header: "h3", animated: false});});', 'inline');
}