kay9 is the winner. I had been looking for a solution to this problem for a few hours, and this was one of the few suggestions that could be tried on a live site. Changing the theme is never really an option on a live site, at least not one that belongs to a client.
Adding this
define( 'WPCOM_API_KEY', 'abcd1234efgh' );
to the bottom of the wp-config.php file did the trick (obviously using my WP API key instead of abcd1234efgh).
Thanks Kay!