How do I register a stylesheet inside a WordPress widget?

wp_enqueue_scripts is called way before WordPress processes the widget content, so just as in this post, you’ve missed the boat :).

Instead, just call wp_enqueue_style directly:

function widget($args, $instance) {
    wp_enqueue_style( 'myprefix-style', plugins_url('mystyle.css', __FILE__) );

    //Widget content
}

(no need to register if you’re just going to enqueue it straight after). Same works for wp_enqueue_script.