Yeah, everthing went smoothely. We declared the emergency and ATC gave us direct to JAN and we were able to descend from 38000 all the way to the ground with no delays at all. ATC took our passenger's information from us and passed it on to the paramedics on the ground. When we pulled up to the gate (no more than 10-15 mins passed from the time we declared the emergency and the time we were on the ground taxiing in) there was an ambulance, rescue truck, fire chief, and 2 police squad cars waiting at the gate. The instant we popped the door open, they all rushed on board to take care of our passenger.
Everything all worked out in the end (I believe she lived). Only real hiccup was the flight attendants and a nurse on board (great how things work out, isn't it?) used our special emergency medical kit (kind of like a super first aid kit on steriods, and it even has steriods in it) and our emergency oxygen bottles. On the ground, we had to have a mechanic come in from home to refill our oxygen bottles as they are required to be full for every flight (for just this reason)