What you produce at the end is an application that runs under Laravel. I’m not familiar with Laravel specifically, but frameworks generally allow you to run several applications under just one installation of the framework. Like Wordpress, Laravel is designed to work on hosted services, so installing it on a hosted service shouldn’t be a problem nor should the response time be unacceptable.
If you search Google for [php login system], you should get lots of hits.
It didn’t seem like your original question was answered, though. A common way to implement a login system is to use PHP sessions (the $_SESSION variable.
First, you have a login form, basically user id and password. When someone submits the form, you check the user database to see if there’s a row that matches what was entered. If there was, you set a session variable.
$_SESSION['user'] = $login;
Anything like that will work. You may elect to save the user’s name or just set something to TRUE to know someone is logged in.
You also want a link to logout with code that unsets the session variable you set, after which you can use the PHP header statement to redirect the user to the login page.
Then, on every page in your project that shouldn’t be accessed without logging in, the very first lines should be:
session_start();
if ($_SESSION['user'] == FALSE) {
header('Location: http://www.example.com/login.php');
}
On your login page, you start the session but of course don’t check to see if nobody’s logged in or you’ll get an endless redirect. But that’s pretty much it. Obivously, you can also set up form to register users, retrieve passwords, etc. but the above is the basics of how it works. Hope that helps. 
