What country are you flying to Frankfurt from? Is it hard to fly to anywhere else in Germany from your home? Have you been to Germany before?
I personally would not want to SO MUCH tedious driving, on crowded autobahns. You could rent a car after getting near your family destination in Denmark, for example. I guess it is too late to price a "Multiple Cities" ticket into Frankfurt and out of Copenhagen (or Hamburg, if you are still saddled with a Germany-rented car to return.) I would avoid major cities if I had a car in Germany. My wife and I have made many trips where we only visited major cities, by train.
It's certainly possible to go back-track to a car by train after taking a dayboat on the Rhine, but it's not an area where having a car is essential - especially for the short time you will be there. I simply mean that you are going to be focusing on easy-to-get-to places along a river that has train lines on both sides. Cologne, if you go there, is good for 2-3 days alone, if you like a variety of museums and historic sites by walking.
It is entirely convenient to take a single train from northern German cities to, say, Copenhagen.