How to display a widget available for all themes

Put all your custom widget code into a separate php file, add a plugin header:

<?php
/**
 * Plugin Name: Name Of The Plugin
 * Plugin URI: http://URI_Of_Page_Describing_Plugin_and_Updates
 * Description: A brief description of the Plugin.
 * Version: The Plugin's Version Number, e.g.: 1.0
 * Author: Name Of The Plugin Author
 * Author URI: http://URI_Of_The_Plugin_Author
 * License: A "Slug" license name e.g. GPL2
 */

then upload it to your plugins directory and activate it. That’s it!