Editing a pre-built contact form

I am using a theme called MediCenter for the web site http://www.superslowla.com. It includes a form that’s hard-coded in their contact_form.php to be used for doctor appointments, and I need to change field names to fit my use. I created a new contact_form.php script in my child theme to protect the original and edited that file. Apparently I have not revised the form correctly because it neither processes the PHP script nor returns an error. Can someone take a look and see if there is an obvious problem?