Two years ago, we were staying at the Dolphin. Both my kids got sick at different times, first my son late in the evening. He had a fever and was feeling very bad - I called the front desk, and they gave me the number of a clinic that sends the doctors to your room.
Two doctors came to our room (one was training to work at the clinic, but they were both docs). They brought all kinds of tests that one would assume would be common problems (had to test him for the flu as there was concern everywhere about that avian flu). He had an ear infection, which was funny, he hadn't had one in years. Also he was wheezing (he has asthma). They gave me the antibiotics and were in and out in little over an hour.
It was a Godsend! They gave me all the forms to turn into our insurance company (I think the bill was about $400 total, but some of it was an extra $50 or so for it being after midnight). I couldn't believe that my insurance company, and HMO, reimbursed us for everything!
Then two days later, DD is running a fever (really high also). I called the same clinic, and she had strep throat. Same great service!
If you need that, just ask the front desk, they go to all the Disney hotels. It was way better than driving them somewhere (with fevers and DD vomited also) plus I didn't have a car.
Even if I wasn't reimbursed, it would have been worth it. In the few times we've had to use the ER here at home, it always takes them a while, plus this way the kids could be resting in bed while we waited.