Be careful with hotels in the Hauptbahnhof area in Stuttgart right now. The train station is in the process of a massive and messy reconstruction (cost is above 4 billion euros) and the situation in this area is, let's say, interesting...

If you are in Kernen (not Kerner), then staying in Stuttgart downtown is not the best strategy anyway, since the commute to Kernen will not be easy. I'd rather take a hotel in Waiblingen or in Kernen and take the bus or a taxi. The typical www-pages identify a few good hotels there. Transportation to Kernen is so-so, from Stuttgart you'd have to take the subway to Waiblingen and then the bus. With a rental car, stay in Waiblingen, Kernen, or Esslingen, or one of the smaller villages in between, and avoid Stuttgart (and Fellbach), as the traffic can be difficult.