When you see "Invalid domain for site key", then the key is okay in general, but not for your domain. The server can not test this case, so an effective lockout prevention is not possible.
You will either need one of the following: - access to the settings for your sitekey on
reCaptcha API key administration - access to your WordPress installation (via SSH or FTP) or database access - database access
With API key admin
Look at source code of the login page.
Find the part saying data-sitekey="XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX" (The XXX-part should be your sitekey.)
Go to the Google reCaptcha API key administration
Find the list entry with the sitekey from step 2
If lockout prevention is enabled you can simply delete the key set up a new one. If not enter your domain name at "Domains" in a new line and wait up to 30 minutes.
With FTP Access:
Add this line of Code somewhere at the end of your theme functions.php:add_filter('wp_recaptcha_required','__return_false');
This will disable the chaptcha everywhere.
Set up a new keypair and test it.
Remove the line above from your theme functions.php.
If you have Database access
Execute the following SQL-Commands in your Database: DELETE FROM wp_options WHERE option_name = 'recaptcha_publickey'; DELETE FROM wp_options WHERE option_name = 'recaptcha_privatekey';
(Please note that wp_options might have a different prefix in your installation.)
After the login you will see a message asking you to set up the API keys.
Set up a new keypair on Google and test it.
If none of these works for you
That's too bad...