Validation not working in custom form

I have a custom form. I have gotten the CAPTCHA to display, but no matter where I enter the CAPTCHA validation lines to the section of validation code, it breaks the validation for the form and also does not try to validate the CAPTCHA. I even had the developer of my theme try to place the lines and they could not figure it out. I have read all of the forum questions and see that anytime someone is asking this question, the answer is a copy-paste of the generic instructions. Can someone please assist with specific info about placing of the validation code?

Here is the validation code from my form:

<?php
                $subject_error_msg = get_option('contact_subject_error_message');
                $name_error_msg = get_option('contact_name_error_message');
                $email_error_msg = get_option('contact_email_error_message');
                $message_error_msg = get_option('contact_message_error_message');
                $mail_success_msg = get_option('contact_message_successful');
                $mail_error_msg = get_option('contact_message_unsuccessful');

                ?>

                <!-- Validate script -->

                <script type="text/javascript">
                    function validate_email(field,alerttxt)
                    {
                        with (field)
                        {
                            apos=value.indexOf("@");
                            dotpos=value.lastIndexOf(".");
                            if (apos<1||dotpos-apos<2)
                            {$('#contact-error').empty().append(alerttxt);return false;}
                            else {return true;}
                        }
                    }

                    function check_field(field,alerttxt,checktext){
                        with (field)
                        {
                            var checkfalse = 0;
                            if(field.value == ""){
                                $('#contact-error').empty().append(alerttxt);
                                field.focus();checkfalse=1;}

                            if(field.value==checktext)
                            {
                                $('#contact-error').empty().append(alerttxt);
                                field.focus();checkfalse=1;}

                            if(checkfalse==1){return false;}else{return true;}
                        }
                    }

                    function checkForm(thisform)
                    {
                        with (thisform)
                        {

                            var error = 0;

                            var message = document.getElementById('message');
                            if(check_field(message,"<?php echo $message_error_msg?>","")==false){
                                error = 1;
                            }

                            var email = document.getElementById('email');
                            if (validate_email(email,"<?php echo $email_error_msg?>")==false)
                            {email.focus();error = 1;}

                            var contactname = document.getElementById('contactname');
                            if(check_field(contactname,"<?php echo $name_error_msg?>","")==false){
                                error = 1;
                            }

                            if(error == 0){
                                var contactname = document.getElementById('contactname').value;
                                var email = document.getElementById('email').value;
                                var message = document.getElementById('message').value;

                                return true;
                            }

                            return false;

                        }

                    }

                </script>

                <!-- end of script -->

https://www.ads-software.com/plugins/captcha/