nemonem

Solved the problem of search box appearing in each menu with:

<?php add_filter('wp_nav_menu_items','add_search_box', 10, 2); 

function add_search_box($items, $args) {
  if ($args->theme_location == 'primary_navigation') {
        ob_start();
        get_search_form();
        $searchform = ob_get_contents();
        ob_end_clean(); 

    $items .= '<li class="searchbox">' . $searchform . '</li>';
  }
  return $items;    

}?>

My inelegant solution for styling:

<?php add_filter('wp_nav_menu_items','add_search_box', 10, 2); 

function add_search_box($items, $args) {
  if ($args->theme_location == 'primary_navigation') {
        ob_start();
        get_search_form();
        $searchform = ob_get_contents();
        ob_end_clean(); 

    $items .= '<li class="menu-search-icon-kt panel"><a class="menu-search-btn collapsed" data-toggle="collapse" data-parent="#menu-main-menu" data-target="#kad-menu-search-popup" style="height: 120px; line-height: 120px;"><i class="icon-search"></i></a><div id="kad-menu-search-popup" class="collapse"><form role="search" method="get" id="searchform" class="form-search" action="https://MyWebsiteName.com/"> <label class="hide" for="s">Search for:</label> <input type="text" value="" name="s" id="s" class="search-query" placeholder="Search"> <button type="submit" id="searchsubmit" class="search-icon"><i class="icon-search"></i></button></form></div></li>';
  }
  return $items;    

}?>

Clearly there is a better way and I need to improve my understanding of PHP and CSS, but it is working for now.