Recaptcha for hard-coded custom contact form

The web developer who made my website hard-coded the custom contact form into the website page so I could not find the slug needed to follow this instruction.

What I did was I installed the plugin, activated it, etc. and then inserted the following code (found on Google Recaptcha documentation) just before the SUBMIT button.

<script src="https://www.google.com/recaptcha/api.js?render=_reCAPTCHA_site_key"></script>
<script>
grecaptcha.ready(function() {
    grecaptcha.execute('_reCAPTCHA_site_key_', {action: 'homepage'}).then(function(token) {
       ...
    });
});
</script>

Is there any better way to implement Recaptcha to hard-coded custom contact form? Thanks in advance