How do I get a data value after all AJAX calls are complete?
Seems that you are building the elements of the form some parts using ajax and some parts using what you show on JsFiddle.

To run commands after success of an Ajax callback you could use he Ajax framework and Ajax framework commands API documentation. Here is a brief example. You will the hook_form_alter to alter the linkit_dashboard_form and add the commands. This alternative is more accurate to execute the commands after a specific ajax call but would require many changes adding hooks and custom functions.

Another alternative is to use .ajaxComplete() and check if the ajax call retrieves the linkit form elements. Example:

$(document).ajaxComplete(function(event,request, setting) {
 // Add same check you use on your setTimeout to make sure that "form-item-linkit-rel" input exists. 
 // You could also use some of the values of the "setting" param of ajaxComplete() function to use additional checks.
});


Note: In case that the problem is caused by something declared inside of  Drupal.behaviors.mymoduleLinkit, try to debug your Js code with Chrome/Firefox DevTools. Add some breakpoints to find out what is causing the issue.