Firstly, thanks to linking to my blog post :)
Secondly, what mbrsolution was suggesting was to completely replace all of the WordPress files with a fresh copy of WordPress. To do this, do the following:
1. Log into your server via FTP and download all of the wp-* files and folders, so that you have a backup of your site.
2. Delete all of the wp-* files and folders.
3. Download a fresh copy of WordPress
4. Extract the zip file and upload it to your server in the same place as your previous installation
5. Copy the wp-config.php file from your backup
This should get you up and running with a clean installation of WordPress, with no plugins or themes. Hopefully you will then be able to log in.
I tried accessing your login page and I too saw the Google Authenticator box, so it looks like a problem on your server. Deleting the folder containing the Google Authenticator plugin should be enough to get rid of it normally though.
The other thing to try is restoring a backup of your phone from before you installed the app update. If you have iCloud backups turned on or sync regularly with iTunes, then you should be able to take your phone back a couple of days and hopefully get the old app working again with your existing tokens.
I was fortunate that I still had the login cookies on my site so I was able to get in without any problems when I updated.