From Anchorage, probably on the Penair flight, it's either non-stop or one-stop at St. George. It's only 15k miles using Alaska miles, or very expensive using cash. I'm not aware of any other scheduled service.

I've been wanting to get out there, but won't get a chance this summer. I would like to hear where you are staying and the cost as well. I thought about camping, but the winds and weather would most likely make that unpleasant. The only hotel I know of, if it exists, is the King Eider Hotel.