Thank you for all of these great replies and recommendations. Because this was tied in with a work trip, it made it really hard, in the end, to make most of these ideas feasible but they're still great.
A few notes of what I did see: I passed through Wuppertal on the regional train and saw their suspended train over the river. That was incredible and I'd go back to see more of it.
Neheim and Arnsberg were really charming towns with lots to see around them. I stayed at the Dorint in Neheim. It's a business hotel, but there's a very charming old lodge and inn next door called Rodelhaus which serves traditional and very good food. Spargel was in season.
The area is a bit of a recreational destination. There's hiking around Neheim and old ruins situated within the Arnsberg Forest, and the towns are part of an east/ west biking trail that extends through the Saarland. There's a lake to the north, and I was told people visit the area around the Dorint hotel for sledding in the winter. There are also several caves with tours nearby, and breweries like Veltins and Grevensteiner offer tours.
From Arnsberg, I took the regional Saarland train east to Kassel to connect to an ICE to Frankfurt. A day in Frankfurt turned out to be more feasible for me before my flight. That said, if I'd had another week, I think I could have easily used it up throughout this region. Cologne was great to visit and it seems like a great jumping off point to many other cities and towns in Westphalia.
Thank you all for the great recommendations.