Need Budget, Pet Accepted Accommodations Ataturk Airport, Istanbul, Turkey

I am accepting a Teach English Abroad assignment in Odessa, Ukraine with a start date the second week of April, 2011. As this may be a long-term position, I'm bringing my pet cat with me. I am taking Turkish Air through Ataturk International. Does anyone know of a very budget minded accommodation (Hotel/Motel) that accepts pets? I'm going to be on the ground 20 hours 30 minutes in Istanbul. Thank you!