How to continuously send feedback via AJAX responses to my client?

wp_send_json just does a basic data sanitization under the hood using wp_json_encode function.

So you can use PHP’s built-in functionality json_encode the problem that you’re going to have is keeping the connection alive, If you’re not going to have a lot of visitors you can block the PHP thread by simply running a while true loop and echo data whenever it’s ready.

while(true) {
    // BTW, You'll have to handle your data within the loop or in an outer process
    if($data_available) {
        echo json_encode($data);
        if($last_item) {
            wp_die();
        }
    }
}

You have to make sure you’re sending Keep-Alive header and are tuning your script to provide sufficient requirements for this protocol feature.

Writing this code inside the WordPress’s core is a little bit hard and might get confusing in the future, there is an alternative solution to this problem.

It’s better to use a duplex protocol (like WebSocket) for this kind of data (which are getting ready over the time).
In PHP there is a very good library to write reactive code called ReactPHP, You can simply integrate it with another library called React Socket or Ratchet (I can’t remember correctly if ratchet does handle all stuff by it self or not).