Returning JSON data with ajax in wordpress

Ok so a fairly long question here. I'm fairly new to AJAX and especially using it in the context of WordPress, but I've been following along some tutorials online and I think I'm almost there.

I'll paste what I have so far and explain my thinking.

Ok so to start, the JS.

jQuery(document).ready(function(){
     jQuery('.gadgets-menu').mouseenter(function(){

          doAjaxRequest();
     });
});

Mouse enters .gadgets-menu and the request triggers, using mouseenter so it fires once.

The request itself.

function doAjaxRequest(){
     // here is where the request will happen
     jQuery.ajax({
          url: 'http://www.mysite.com/wp-admin/admin-ajax.php',
          data:{
               'action':'do_ajax',
               'fn':'get_latest_posts',
               'count':5
               },
          dataType: 'JSON',
          success:function(data){
                //Here is what I don't know what to do.                 

                             },
          error: function(errorThrown){
               alert('error');
               console.log(errorThrown);
          }


     });

}

Now the php function.

add_action('wp_ajax_nopriv_do_ajax', 'our_ajax_function');
add_action('wp_ajax_do_ajax', 'our_ajax_function');
function our_ajax_function(){


     switch($_REQUEST['fn']){
          case 'get_latest_posts':
               $output = ajax_get_latest_posts($_REQUEST['count']);
          break;
          default:
              $output = 'No function specified, check your jQuery.ajax() call';
          break;

     }


         $output=json_encode($output);
         if(is_array($output)){
        print_r($output);   
         }
         else{
        echo $output;
         }
         die;
}

And the ajax_get_latest_posts function

function ajax_get_latest_posts($count){
     $posts = get_posts('numberposts='.'&category=20'.$count);

     return $posts;
}

So, if I've done this right the output should be $posts = get_posts('numberposts='.'&category=20'.$count); ie. the number of posts (5), from category 20. I don't know what to do with that now, how do I get the title and the thumbnail?

I'm sorry if this is silly, I'm just fumbling around here.

Amended php

add_action('wp_ajax_nopriv_do_ajax', 'our_ajax_function');
add_action('wp_ajax_do_ajax', 'our_ajax_function');
function our_ajax_function(){


      $output = ajax_get_latest_posts($_REQUEST['count']); // or $_GET['count']
    if($output) {
        echo json_encode(array('success' => true, 'result' => $output));
    }
    else {
        wp_send_json_error(); // {"success":false}
        // Similar to, echo json_encode(array("success" => false));
        // or you can use, something like -
        // echo json_encode(array('success' => false, 'message' => 'Not found!'));
    } 

         $output=json_encode($output);
         if(is_array($output)){
        print_r($output);   
         }
         else{
        echo $output;
         }
         die;
}


function ajax_get_latest_posts($count)
{
    $args = array( 'numberposts' => $count, 'order' => 'DESC','category' => 20 );
    $post = wp_get_recent_posts( $args );
    if( count($post) ) {
        return $post;
    }
    return false;
}

This does not work.

jQuery(document).ready(function(){
     jQuery('.gadgets-menu').mouseenter(function(){

          doAjaxRequest();
     });
});
function doAjaxRequest(){
     // here is where the request will happen
     jQuery.ajax({
          url: 'http://localhost:8888/wp-admin/admin-ajax.php',
          data:{
               'action':'do_ajax',
               'fn':'get_latest_posts',
               'count':5
               },
          dataType: 'JSON',
          success:function(data){
            if(data.success) {
               alert("It works");


                        }
            else {
                // alert(data.message); // or whatever...
            }
        }


     });

}

No alert is shown.

In your code get_posts('numberposts='.'&category=20'.$count); is wrong, but you can use wp_get_recent_posts function instead (though it uses get_posts anyway), for example

function ajax_get_latest_posts($count)
{
    $args = array( 'numberposts' => $count, 'order' => 'DESC','category' => 20 );
    $post = wp_get_recent_posts( $args );
    if( count($post) ) {
        return $post;
    }
    return false;
}

Then in your our_ajax-function you can use

    $output = ajax_get_latest_posts($_REQUEST['count']); // or $_GET['count']
    if($output) {
        echo json_encode(array('success' => true, 'result' => $output));
    }
    else {
        wp_send_json_error(); // {"success":false}
        // Similar to, echo json_encode(array("success" => false));
        // or you can use, something like -
        // echo json_encode(array('success' => false, 'message' => 'Not found!'));
    }

In you success callback function, you can then check

success:function(data){
    if(data.success) {
        // loop the array, and do whatever you want to do
        $.each(data.result, function(key, value){
            // you can use $(this) too
            // console.log($(this)); // check this for debug and get an idea
        });
    }
    else {
        // alert(data.message); // or whatever...
    }
}

You can read here about wp_send_json_error helper function to learn more about helper functions.

Update :

Also remember that, after $output=json_encode($output); the $output is not an array anymore, instead, it's a json string, so is_array($output) will return false but if you use is_array() just before you encode it using $output=json_encode($output); like

if( is_array( $output ) ) {
    $output = json_encode( $output );
}

In this case, is_array( $output ) will return true.

An example/simulation.