Thanks to @eskimo I was able to get this working and figured I share an answer for any of those trying to do the same!
In the ajax-login-script.js
file all I had to do was change this $('form#login_form').on('#submit', function(e)
to this … $('form#login_form').on('submit', function(e)
just removing the #
before calling submit
.
Hope this helps someone!