If you're looking for "storybook Germany", you have to do a little bit more research in NRW. There's plenty of scenic towns and areas, but they don't jump out at you like they might in, let's say, Bavaria, Hessen or Rheinland-Pfalz. It's the most densely populated German state by far, but much of what you see is the result of functional post-war rebuilding and the needs of heavy industry.
Aachen is a good place to start. Once again, largely rebuilt after the war, but there's enough history left (or restored) for at least a day of sight-seeing. Particularly the remains of Charlemagne's palace and cathedral complex.
Give nearby Stolberg a quick look. It has a nice little preserved historic core that looks very atypical for Germany. Much more similar to the neighboring Ardennes region of Belgium. It has an intact castle, but I don't remember if you can tour it.
I had to cancel a trip there, but I always wanted to give Monschau a look. It's billed as the gateway to Eifel National Park.
Most of the major industrial cities in NRW are pretty interchangeable, but if you want to give some of them a look, I would give priority to Bonn (virtually untouched in the war, maintains it's late 19th century elegance), Köln (modern city, but it's most important historical buildings survived), and Münster (just a nice town).
The more scenic, typically "storybook" part of the state is the semi-mountainous Sauerland region to the east. Siegen is a typical example.
10 days also gives you more than enough time to head a little further south to the famous Mittelrhein. The scenic portions begin a little south of Bonn, and continue to Bingen, with the most popular scenery upstream of Koblenz.
