Reading your explanation in the comments section of your question I believe Josh has hit upon the answer.
The reason your jQuery event is not working is because, you "attach" the event handlers using $('#delete').click(function() {alert('success');})
to the anchor/button with id "delete" (that's the #delete part), then later that anchor "#delete" get's removed from the DOM when you replace the innerHTML of "#NewsFeedListID", so you would either have to re-attach the event handlers to #delete, or as Josh has suggested use a technique known as event delegation to keep the click event triggering even though you are replacing the innerHTML.
I'll leave it to you to look into "event delegation" and using jQuery to simplify your ajax requests (look into the method "load" for a very simple alternative to what you are describing) but use $('#delete').live('click',function() {alert('success');})
to solve your immediate problem.