I don't have suggestions for a hotel, but you could also stay in Kusadasi a little longer and have a leisurly morning/afternoon then take a private shuttle to the Izmir airport later in the day for your overnight. I used Welcome Pickups to do something similar for an afternoon flight. There are other shuttle services as well, including through the hotel.
If you take the tour bus to the Izmir airport, there isn't much right around the airport, but I believe it's pretty easy to get to the city center by public transit. Last time I checked there were no left luggage services at Izmir airport, so you'd want to be sure you could leave your bags with a hotel if you head off to sightsee. Depending on where you fly to next, give yourself a little time at the airport. The international terminal is very small, but you have to go through security right at the door, then again after you check in with the airline. But it's possible to arrive too early - we arrived 3 hours early for our flight and had to wait for the airline desk to open so we could check our bags.