Any page should only have one set of <html>, <head>, and <body> tags. Adding that code doubles up those tags. I’m not sure if you can have more than one ` tags either.
You might try searching on adding reCAPTCHA to an existing form, as that code is for a simple demo page (hence the html, head, and body tags)
There seems to be other problems on your site, like missing styling.
Frankly, if you’re using your site to promote a business, I’d recommend upgrading to Backlight, which integrates reCAPTCHA among a ton of other things.
Hi Reid, reCAPTCHA also requires changes to the PHP that verifies that the reCAPTCHA has been correctly completed. I noticed that your admin page is coming up with a PHP error. It looks like your host has upgraded PHP and your old TTG code is no longer compatible with it. You could try changing the version of PHP to an older version, if your control panel allows, or look at upgrading to Backlight.
I think I get charged extra by my ISP for using an older version of PHP, so time to upgrade or switch to a Wordpress photography theme since my business is more focused on teaching than photographing.