Good luck finding one. I am searching for a room for the Florida weekend, and EVERYTHING even remotely close is booked. The closest I have found is a few rooms at the Cedar Bluff exit, west of town (12ish miles). I actually made a safety booking at an airport hotel. It was way better than some of the flea bags (Ramada @ West Town I'm looking in your direction) that were still available in West Knoxville. I have never stayed near the airport, but it is a straight shot, about 12 miles, down Alcoa Highway. Not too bad.
A couple of the places still available out there were fairly new, and pretty decent looking. Not ideal, but better than nothing. My girlfriend reminded that it could be worse...I could be stuck in Nashville that weekend.