Hi there! Not sure where you are flying from . . . you may want to overnight in Fairbanks for the jet lag - Denali Park is about 2 1/2 hrs from Fairbanks, so you wouldn't show up until about 11. I would highly recommend that you take a shuttle bus into the park - to Wonder Lake or at least Fish Creek. That would take up most of your Saturday. Sunday morning you could take in a doggie demo, do a raft trip or something, and then head back to Fairbanks, maybe stop at the museum at the University and see a bit of Fairbanks before your flight.
Lots of hotels in the Denali Park area, as well as a few hotels, lots of cabins and B&B's in the Healy area. Here is our local chamber page which may give you some ideas - everything from the very basic to the fairly upscale (for Alaska). http://www.denalichamber.com/ If you want specific recommendations, I would be happy to help. Good luck! :)