Wow, this is quite the pickle. The good news is that you are correct that justinchina.co.uk is causing your issues; the bad news is that it looks like your site's been hacked. Okay, first take a deep breath and have a hug. I've been there, and it sucks. But let's work through this. (I'm going to link to some resources as I'm not sure where you are in your SEO knowledge journey.)
Right now, your site www.petmedicalcenter.com is being 301 redirected (a permanent redirect and the source of why your rankings dropped) to justinchina.co.uk, which is giving a 200 or okay code. (Google is probably a bit confused as your WordPress SEO plug-in is showing www.petmedicalcenter.com as the rel=canonical page, which is in your favor.)
First, to completely verify that you've been hacked, I need you to go to your .htaccess file. This file is located at http://www.petmedicalcenter.com/.htaccess and currently blocked from my outsider view. I'm pretty certain in there, you'll see the code that's 301'ing it to justinchina.co.uk. You'll need to remove this code, just don't do it yet as we need to stop the hacker first. Since you've been able to log into the WordPress, the hacker hasn't change your passwords and such. (I highly recommend backing up your entire site so you have a copy before you do anything else!)
When I was digging into your site, I believe that your GoDaddy account was hacked and the 301 redirect was created that way, instead of your site being hacked. The reason I believe this is is because the IP on both your site and the duplicate are the same. This means that whomever is behind justinchina.co.uk has taken control of your domain too. I've attached screenshots, but you can look at the live data: http://who.is/whois/justinchina.co.uk and http://who.is/whois/petmedicalcenter.com. It's highly likely that the information for JustinChina is fake information.
First, you need to see if you can log into GoDaddy using your current credentials. If you can, you need to go into your account information and verify that your information is correct (email address, billing address, phone, etc.). If it not, you need to update it before you start doing anything else. (Don't send your password reset to the hacker's email!) Then you need to change your GoDaddy password. Next you need to contact GoDaddy and get your domain back. (I assume you also hosted with GoDaddy since that's where it's registered through.)
Where you might run into a problem is if your GoDaddy log-in doesn't work. Then you need to go directly to GoDaddy and give them all the information about who you are and what's happened to your site. If you are in this situation, you are at the mercy of GoDaddy's customer support to get your domain back.
Once GoDaddy gives you control back of your domain, then make sure to change all your WordPress or other domain-related passwords. (If you are using your GoDaddy log-in password for anything else, make sure to change it too!) Then go and remove that 301 redirect in the .htaccess file. And then make sure your WordPress, WordPress plug-ins, and WordPress themes are completely updated.
I took a look at your site's code, and I couldn't see any malicious code inserted into your site. The good news is that it looks like a pretty clean hack into your GoDaddy account and a 301 redirect, so once you have your domain back, there's less to do. But I do recommend going over your code with a fine-tooth comb, just in case.
Your rankings should recover over time. Unfortunately, it will just take time, but you can use it as a benchmark that everything's going good with your site.
I hope this helps you start cleaning up this mess and wish you the very best on sorting this out. Please let us know how it goes.
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