How do I get my custom widget to show on live site?

A widget class object contains four functions with mandatory names:

  1. __construct(), which creates the widget object
  2. form ($intance), which generates the form in the admin
  3. update ($new_instance, $old_instance), which saves the form data in the admin
  4. widget ($args, $instance), which displays the widget on the front end.

Your nr 4, widget, is empty except for an extract statement. Nothing is echoed. So nothing shows on the front end.