WordPress widget / plugin….fields not appearing in my widget

With your code given in OP, it is hard to exactly tell you where you are going wrong. One note on what I can, you should never use create_function(), it is a security thread that can be exploited by hackers. You should avoid using that completely. Rather make use of proper closures that was introduced in PHP 5.3.

To help you, here is a basic skeleton which you can use to build a widget with. Just one note, this requires PHP 5.4 due the the use of the new short array syntax ([])

class Custom_Services extends WP_Widget 
{

    public function __construct() 
    {
        parent::__construct(
            'widget_custom_service', 
            _x( 'Custom Service Widget', 'Custom Service Widget' ), 
            [ 'description' => __( 'Displays information from a custom service.' ) ] 
        );
        $this->alt_option_name="widget_custom_service";

        add_action( 'save_post', [$this, 'flush_widget_cache'] );
        add_action( 'deleted_post', [$this, 'flush_widget_cache'] );
        add_action( 'switch_theme', [$this, 'flush_widget_cache'] );
    }

    public function widget( $args, $instance ) 
    {
        $cache = [];
        if ( ! $this->is_preview() ) {
            $cache = wp_cache_get( 'widget_services', 'widget' );
        }

        if ( ! is_array( $cache ) ) {
            $cache = [];
        }

        if ( ! isset( $args['widget_id'] ) ) {
            $args['widget_id'] = $this->id;
        }

        if ( isset( $cache[ $args['widget_id'] ] ) ) {
            echo $cache[ $args['widget_id'] ];
            return;
        }

        ob_start();

        $title          = ( ! empty( $instance['title'] ) ) ? $instance['title'] : __( 'Category Posts' );
        /** This filter is documented in wp-includes/default-widgets.php */
        $title          = apply_filters( 'widget_title', $title, $instance, $this->id_base );

        // ADD YOUR CUSTOM PHP CODE HERE FOR EXECUTION TO DISPLAY ON FRONT END

        echo $args['after_widget']; 

        if ( ! $this->is_preview() ) {
            $cache[ $args['widget_id'] ] = ob_get_flush();
            wp_cache_set( 'widget_services', $cache, 'widget' );
        } else {
            ob_end_flush();
        }
    }

    public function update( $new_instance, $old_instance ) 
    {
        $instance                   = $old_instance;
        $instance['title']          = strip_tags( $new_instance['title'] );
        $this->flush_widget_cache();

        $alloptions = wp_cache_get( 'alloptions', 'options' );
        if ( isset($alloptions['widget_custom_service']) )
            delete_option('widget_custom_service');

        return $instance;
    }

    public function flush_widget_cache() 
    {
        wp_cache_delete('widget_services', 'widget');
    }

    public function form( $instance ) 
    {

        $title      = isset( $instance['title'] ) ? esc_attr( $instance['title'] ) : '';
        ?>

        <p>
            <label for="<?php echo $this->get_field_id( 'title' ); ?>"><?php _e( 'Title:' ); ?></label>
            <input class="widefat" id="<?php echo $this->get_field_id( 'title' ); ?>" name="<?php echo $this->get_field_name( 'title' ); ?>" type="text" value="<?php echo $title; ?>" />
        </p>

    <?php
    }

}

add_action( 'widgets_init', function () 
{
    register_widget( 'Custom_Services' );
});