How does one include the JQuery UI plugin to wordpress and enque it correctly in a plugin form?

It is important, that you include your script with wp_enqueue_script() and defined the scripts, there are use your script. WP control the scripts. Your example has the problem, that your div and js inlcude this load before load jquiry and ui. Also it is possible, that WP load al scripts in Footer and then have you also the same problem.
Hint: http://wpengineer.com/2028/small-tips-using-wordpress-and-jquery/