What to do with 6 nights in Cologne?

How many people, ages, time of year, car or not, personal interests?

Cologne is good for at least 3 days by itself, but except for Dusseldorf isn't that close to major attractions. You can take the train to Bingen and the KD boat to Bacharach or St. Goar, and train back to Cologne. But that is likely to be a full day.

Koblenz is a good day-out. I'd love to see the overhead subway in Wuppertal, but unless the home dance company is performing, I can't think of any other reason to go there! Have you see the movie, The Princess and the Warrior?

Since you have the free hotel stay - might I suggest being open to one over-night "day' trip if there is somewhere that speaks to you. For example, while staying in a apartment in Florence, I splurged on a night in a nearby beach town that made a very nice one night trip but would have been an exhausting trip if we had to return that night. Just in case there is someplace others recommend like that (and leave your stuff at hotel and just bring a day bag)

If you have a car, Aachen and Maastricht are just over an hour away. The Mosel valley is a little far for a day trip, but it would be a great change from the city to spend a night in one of the towns on the river.

Suggest looking for attractions and museums over on Trip Advisor for both Cologne and Dusseldorf.

Both Aachen and Maastricht are also very easy by train.

Try the Schwebebahn in Wuppertal. Not far away.

I just returned from time there. If you have a nice weather day, a good day trip and time on the river could be to take the train to Bonn, then the KD boat to Linz. That part of the river is very scenic. Linz is a very attractive town full of half timbered buildings. We took the boat back to Bonn because it was such a beautiful day, but you could easily just take the train back to Cologne. We talked about taking the boat to Koblenz, but it would have made for a very long day.