Hierarchical list of menuitems

I’m trying to make a list of menuitems based on the page I’m viewing. For example if I’m on the page History, I have a menu with all the subcategories under history and the posts in these subcategories. Also the menuitems have to be clickable.

So basically I want this:

MAIN CAT = pagename

SUB CAT1
– post1
– post2

SUB CAT2
– post1
– post2

This is what I hacked together so far:

<div id="menu">
<ul>
    <?php
    $page_title = wp_title();
    preg_replace( "/[^a-z0-9 ]/i", "", $page_title);
    strtolower($page_title);

    $cat_id = get_cat_ID($page_title);
    //get terms (e.g. categories or post tags), then display all posts in each retrieved term
    $taxonomy = 'category';//  e.g. post_tag, category
    $param_type = 'category__in'; //  e.g. tag__in, category__in
    $term_args=array(
      'orderby' => 'name',
      'order' => 'ASC',
      'child_of' => '$cat_id'
    );
    $terms = get_terms($taxonomy,$term_args);
    if ($terms) {
      foreach( $terms as $term ) {
        $args=array(
          "$param_type" => array($term->term_id),
          'post_type' => 'post',
          'post_status' => 'publish',
          'showposts' => -1,
          'ignore_sticky_posts'=> 1
          );
        $my_query = null;
        $my_query = new WP_Query($args);
        if( $my_query->have_posts() ) {

          echo '<li><a href="' . get_category_link( $term->term_id ) . '" title="' . sprintf( __( "View all posts in %s" ), $term->name ) . '" ' . '>' . $term->name. '</a> ';

          while ($my_query->have_posts()) : $my_query->the_post(); ?>
            <ul><li><a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></li></ul>
           <?php
                 endwhile;
        }
      }
    }
    wp_reset_query();  // Restore global post data stomped by the_post().
    ?>
</ul>
</div>

And now it just prints the title of the page.