Even if it doesn't work out for you, this really is an "only in America" type of experience, so it will be a win regardless. And if you have a few minutes to surf online, you might look to see if any of the chapters at W&M have alumnae associations in London, beyond Kappa which was shared above. If you can express interest in lifetime membership, that may help you a bit. But admittedly, the college girls don't know much about sorority membership after college
It's big and fulfilling and lasts WAY longer than your time in college. Just a thought.