Get the value attribute of the selected LI element in jQuery UI menu

I have prepared a jsFiddle for my question:

For that I have taken the jQuery UI menu with categories example and have just added VALUE="some_number" attribute to each <LI>-element:

<ul id="menu">
  <li class="ui-widget-header"> Category 1 </li>
  <li value="1"> Option 1 </li>
  <li value="2"> Option 2 </li>
  <li value="3"> Option 3 </li>
  <li class="ui-widget-header"> Category 2 </li>
  <li value="4"> Option 4 </li>
  <li value="5"> Option 5 </li>
  <li value="6"> Option 6 </li>
</ul>

Then I am trying to retrieve and display that value in an alert on a button click:

$("#menu").menu({
  items: '> :not(.ui-widget-header)'
});

$('#btn').button().click(function(ev) {
  ev.preventDefault();
  var value = $('#menu').val();
  // var value = $('#menu li').attr('value');
  alert('Selected menu value: ' + value);
});

but the .val() seems to be a bad choice here, I probably need to go through $("#menu") for that?

Also I wonder, why are the list items highlighted on hover and on item selection in the jQuery example - but not in my jsFiddle?

UPDATE:

The HTML-select/optgroup/option workaround suggested by clearshot66 is nice, but I would like to get my jQuery UI menu working... I have also posted my problem at the jQuery forum.

Try something similar to this, it'll do what you're looking for a little cleaner

Also note, your hover isn't working because you need to add a hover attribute on your CSS

Example for yours, not mine: #menu li:hover{background-color:yellow;}

$('#btn').click(function() {
  var value = $('#menu').val();
  alert('Selected menu value: ' + value);
});

#menu{overflow:auto;}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<select id="menu" size='6'>
  <optgroup label="Swedish Cars">
    <option value="volvo">Volvo</option>
    <option value="saab">Saab</option>
  </optgroup>
  <optgroup label="German Cars">
    <option value="mercedes">Mercedes</option>
    <option value="audi">Audi</option>
  </optgroup>
</select>

<p><input type="button" id="btn" value="Show Selected" /></p>

Otherwise, for yours, since they're li, you'd need to add the following:

• on hover css to show color change
• on click to show color change, then uncolor the previous selected
• on click event to get the li text, since li don't have value it would be $("#menu li").hasClass("active") basically finding the li that has the color highlight/class active in this case, then getting the .text() value...

In short, a with opt groups would be much more efficient code wise, and can be CSS styled to look like your example as well.