I believe the updated fiddle at http://jsfiddle.net/TxYh2/3/ shows how to do what you want. The change is as follows:
ui.item.off('click').on('click', 'a', function () {
ui.item.append('<div>Properties for ' + ui.item.text() + '</div>');
});
The main point is to attach the event listener to the specific item that's being added, not the entire document, which you can get using the ui.item reference. This reference also lets you get information about the item being added for your new properties div.