<input type=submit> should be part of a <form>, so you need to submit that form (instead of "pressing" the submit buttton). WWW::Mechanize has two methods for that: submit() and submit_form(). See WWW::Mechanize. Also read WWW::Mechanize::Cookbook and WWW::Mechanize::FAQ.

If the form contains no other input than the submit button (i.e. no other <input> or <button> or <textarea> elements), and the form's method is set to GET, you could also issue a simple GET request ($mech->get(...)) to the form's action URL.

Alexander