How to include second layer sub menu to Walker Nav?

Without testing & debugging your example I’m not 100% certain what’s going wrong. I noticed your walker doesn’t have a end_lvl method which would be my first guess as to why dropdowns past the first level won’t work.

Here’s a copy of a Nav Walker that I’m certain handles nested nav menus well. You’ll want to replace the start_el contents with those from your example.

<?php
class Walker_Nav_Primary extends Walker_Nav_Menu {


    function start_lvl( &$output, $depth = 0, $args = array() ) {
        $indent = str_repeat("\t", $depth);
        $output .= "\n<ul class=\"sub-menu\">\n";
    }

    function end_lvl( &$output, $depth = 0, $args = array() ) {
        $indent = str_repeat("\t", $depth);
        $output .= "</ul>\n";
    }

    function start_el( &$output, $item, $depth = 0, $args = array(), $id = 0 ) {
        $indent = ( $depth ) ? str_repeat( "\t", $depth ) : '';

        $class_names = $value="";

        $classes = empty( $item->classes ) ? array() : (array) $item->classes;
        $classes[] = 'menu-item-' . $item->ID;

        $class_names = join( ' ', apply_filters( 'nav_menu_css_class', array_filter( $classes ), $item, $args ) );

        $class_names = $class_names ? ' class="' . esc_attr( $class_names ) . '"' : '';

        $id = apply_filters( 'nav_menu_item_id', 'menu-item-'. $item->ID, $item, $args );
        $id = $id ? ' id="' . esc_attr( $id ) . '"' : '';

        $output .= $indent . '';

        $attributes  = ! empty( $item->attr_title ) ? ' title="'  . esc_attr( $item->attr_title ) .'"' : '';
        $attributes .= ! empty( $item->target )     ? ' target="' . esc_attr( $item->target     ) .'"' : '';
        $attributes .= ! empty( $item->xfn )        ? ' rel="'    . esc_attr( $item->xfn        ) .'"' : '';
        $attributes .= ! empty( $item->url )        ? ' href="'   . esc_attr( $item->url        ) .'"' : '';

        $item_output = $args->before;
        $item_output .= '<li'. $class_names .'><a'. $attributes .'>';
        $item_output .= $args->link_before . apply_filters( 'the_title', $item->title, $item->ID ) . $args->link_after;
        $item_output .= '</a>';
        $item_output .= $args->after;

        $output .= apply_filters( 'walker_nav_menu_start_el', $item_output, $item, $depth, $args );
    }


    function end_el( &$output, $item, $depth = 0, $args = array() ) {
        $output .= "\n";
    }
}