I'm just wondering is there a specific link or piece of code I can use to create a launch button for launching the website i.e taking it out of maintenance mode?
You could find the flag in the database which is either 1 or 0 depending on whether maintenance mode is enabled or not.
You could then write a script and put it on a page which allows a button press to change the flag status in the database, thus putting the site online.
i.e.
PHP page (presumably behind a password wall) which contains a form with a submit button to a PHP script, which contains a mySQL UPDATE query to change the flag status.