PHP contact form seems to be unresponsive

Question

Here is my project I'm working on: http://www.designbynoemi.co.uk. In the contact section the html/php contact form doesn't seem to be working. Php is running on the server. No error messages showing up.

I would appreciate any sort of directions. Thanks!

Here is my html markup:

    <form id="contact-form" class="contact-form" action="contact.php" method="POST">
            <p class="contact-name">
                <input id="contact_name" type="text" placeholder="Full Name" value="" name="name" />
            </p>
            <p class="contact-email">
                <input id="contact_email" type="text" placeholder="Email Address" value="" name="email" />
            </p>
            <p class="contact-message">
                <textarea id="contact_message" placeholder="Your Message" name="message" rows="15" cols="40"></textarea>
            </p>
            <p class="contact-submit">
                <a id="contact-submit" class="submit" href="mailto:[email protected]">Send Your Email</a>
            </p>

            <div id="response">

            </div>
    </form>

And my PHP code:

    <?php

header('Cache-Control: no-cache, must-revalidate');
header('Expires: Mon, 26 Jul 1997 05:00:00 GMT');
header('Content-type: application/json');

$admin_email = '[email protected]'; // Your Email
$message_min_length = 5; // Min Message Length


class Contact_Form{
    function __construct($details, $email_admin, $message_min_length){

        $this->name = stripslashes($details['name']);
        $this->email = trim($details['email']);
        $this->subject = 'Contact from Your Website'; // Subject 
        $this->message = stripslashes($details['message']);

        $this->email_admin = $email_admin;
        $this->message_min_length = $message_min_length;

        $this->response_status = 1;
        $this->response_html = '';
    }


    private function validateEmail(){
        $regex = '/^([\w-]+(?:\.[\w-]+)*)@((?:[\w-]+\.)*\w[\w-]{0,66})\.([a-z]{2,6}(?:\.[a-z]{2})?)$/i';

        if($this->email == '') { 
            return false;
        } else {
            $string = preg_replace($regex, '', $this->email);
        }

        return empty($string) ? true : false;
    }


    private function validateFields(){
        // Check name
        if(!$this->name)
        {
            $this->response_html .= '<p>Please enter your name</p>';
            $this->response_status = 0;
        }

        // Check email
        if(!$this->email)
        {
            $this->response_html .= '<p>Please enter an e-mail address</p>';
            $this->response_status = 0;
        }

        // Check valid email
        if($this->email && !$this->validateEmail())
        {
            $this->response_html .= '<p>Please enter a valid e-mail address</p>';
            $this->response_status = 0;
        }

        // Check message length
        if(!$this->message || strlen($this->message) < $this->message_min_length)
        {
            $this->response_html .= '<p>Please enter your message. It should have at least '.$this->message_min_length.' characters</p>';
            $this->response_status = 0;
        }
    }


    private function sendEmail(){
        $mail = mail($this->email_admin, $this->subject, $this->message,
             "From: ".$this->name." <".$this->email.">\r\n"
            ."Reply-To: ".$this->email."\r\n"
        ."X-Mailer: PHP/" . phpversion());

        if($mail)
        {
            $this->response_status = 1;
            $this->response_html = '<p>Thank You!</p>';
        }
    }


    function sendRequest(){
        $this->validateFields();
        if($this->response_status)
        {
            $this->sendEmail();
        }

        $response = array();
        $response['status'] = $this->response_status;   
        $response['html'] = $this->response_html;

        echo json_encode($response);
    }
}


$contact_form = new Contact_Form($_POST, $admin_email, $message_min_length);
$contact_form->sendRequest();

?>