The best way to go IMHO is to write your on HTML form and AJAX based form submission. Just analyse the usual login request send by the client when pressing 'log in' (using the NET tab of firebug for example), and mimic it.
Another, easier way is to use the deeplink module, analyse the structure of the deeplink login form page, and copy that form. You can safely paste it in your HTML somewhere, since it is a simple plain POST form submit.
We did this before with a iframe. But that wasn't a satisfying solution. Maybe you could build a page which will submit the username and password to the loginpage of the mendix app.
I think you should be able to write a PHP script where you use the database from the Mendix application you want to login to. When the login is successful, it should bring you to the proper Mendix website.