Yes, the problem is that the PHP isn't being read, but it's not being read because it's not looking in the right place. There is obviously a difference in location between what you posted and what I posted.
While the concept of the solution is straightforward, actually implementing it can be a little more complicated. In essence, the solution is to just point it to the correct location of the script processing file. Here's where it can get a bit more complicated: how you actually point it to the right file. If you want to leave all your files in the location that they are currently in, then you need to modify the action attribute of your form to point to the correct processing file. You can do this manually, or even better, you can take advantage of WordPress to do it for you (somewhat). Here's how you would do that:
action="<?php get_bloginfo('template_directory'); ?>/assets/inc/contact.inc.php"
Another option is to change how you go about displaying your form altogether. Here's the method I use on a site that I recently completed. Here's the link if you want to see it in context. Here's the code so you can see how it is structured:
* Template Name: Volunteer
<div id="content" class="widecolumn">
<?php require('volunteer.inc'); ?>
$value = $_POST['Email'];
$value = strip_tags($value);
$_POST['Email'] = $value;
<?php require('volunteer_form.inc'); ?>
<?php } ?>
<?php get_sidebar(); ?>
<?php get_footer(); ?>
This file is a page template, so I just need to select this template for use with that particular page in WordPress. The reason I can use the two include files without the full path is because I have set up my php.ini file to the location of the include files, and I have set it up so that it is only accessible from the server, not from the website itself. This has the added benefit of being slightly more secure. Also worth noting, it allows the code to be processed by the same php file, which you can see if you look at the source code of the form.
Hopefully this helps give you some ideas on how to get yours working how you want it to work.