As a rule of thumb, places to spend the night are usually close to bus terminals. You can find love motels right behind the Wonju bus terminal. A room typically cost ₩45,000, complete with a flat screen and blazing fast internet.
If travelling with a family the Wonju Tourist Hotel is located in the centre of town and rooms typically start at around 60000 won per night. This hotel does not have floor heating, so in the winter time ask for a second heater.