Code u have used is correct. You just check out the form_number in some site there may be number form_number 2 or 3. For eg. https://adwards.gooogle.com they used 3 form_number upto 3. print the content after the $mech->get and check what your getting. From there you can verify the number of forms they used.

It will work if than if email and password is correct. May be you send the javascriptEnabled => True. Bcoz. in https site they may check these also. For eg. in adwards.google.com they check.

Before you get the content u login manually with username and password and see the script condition based what they entering and pass the key and value with in the submit_form.