My son studied abroad in Copenhagen last spring. We went to visit in April and it was COLD. Not really the temperature, but there was an icy wind blowing a lot of the time. So bring a warm coat, hat, gloves, and a scarf. I just had a spring jacket and it wasn’t enough. As they say in Denmark “there is no bad weather, only bad clothing”.
In addition to the places mentioned already, we enjoyed a canal boat tour. Check into the Copenhagen Card to see if it will be worthwhile purchasing for your stay. It includes all public transportation which was nice. Also, we took a trip to Frederiksborg Castle which was well worth it. From there we went to Helsingor to visit Kronborg castle (the castle where Hamlet is set). It is a quick ferry ride from there to Sweden, which was fun.
One restaurant we ate at was Host (picked by my son). It is pricey, but quite an experience. We did the tasting menu and they brought out 8 (I think) little courses of very different kinds of foods. I think there are only two dinner seatings, so a reservation is absolutely necessary.
If your daughter is in the DIS program, she should have a great experience. I was quite impressed with the program and all of the travel and field trips the students were able to experience. My son really loved it there - so much so that he got himself an internship over the summer so he could stay!