You know, good question....
I'm thinking, yeah, go ahead and send one. Georgetown seems to be all about those that are truly excited to be there. If you intend to matriculate once accepted, go ahead and send it. If it's just a letter of interest, wait and see. You could also call or email Ms. Sullivan and ask, she gave us her phone and email at the interview.
Enthusiasm and a letter of intent never hurt anyone. Badgering adcoms can, but I think you sound socially conscious enough to avoid that. I hope you get the acceptance! Let us know.