We recently did this same trip to Copenhagen and Stockholm for 2 weeks. Left from San Francisco on United using all our points - so we went business class and changed planes at LHR. Nice using the lounges as well. Copenhagen airport is manageable as the train to central Copenhagen is easy to find and use - about 20 minutes to Central Station. There are several hotels around Central station and it is just a block from the old town area. We stayed at the Hotel Alexandra. It has an elevator but no A/C.
We stayed in Copenhagen for 9 days - as we were meeting friends as well as our son and girlfriend. On our own, we would have split the days between the 2 cities. With your situation, I would look for a hotel within the old town area, but there are a lot of cobblestones. Nice areas to explore is the area around the Norreport Station with the Torvehallerne (food market), Botanical Garden, and Rosenborg Castle and park nearby. There were several hotels in the area as well.
Outside of town, we took the train to The Louisiana Museum of Modern Art. We walked from the train station, but I would think a cab would be available. The grounds are easy to walk as is the museum - outstanding views from the restaurant and garden - and we enjoyed the art.
We took the 5 hour train to Stockholm from the airport. Easy to get there from Central Station and there are only 2 tracks at the airport station - one goes to Sweden and the other back to Central Station. However, the tickets we purchased at Central station a few days before, did not match the posted schedule. The passport control people were of no help. We had the 12:16 ticket, but there was no posting of either our train # or our time. We just got on a train to Malmo - 3 stops - and sought help there. Turns out we were on the right train all along.
In Stockholm, we took a cab to our hotel, Miss Clara. We pick hotels for price and appearance, rather than location. Looking back, I would have preferred to stay near the water and near the Grand Hotel. I recommend that area as it is level without construction going on. It is near the Hop-on Hop-off boat, (highly recommend) shopping, Djurgardan Island with the outstanding Vasa Museum, and Gamla Stan (old town). Gamla Stan is all cobblestones and rises to a hill at central square. The walking is fairly easy and the Nobel Museum is on the square, as well as several restaurants. Also recommended is the tour of City Hall, where the Nobel Banquet is held. An extremely interesting building with lovely rooms, though starting and ending with a huge staircase. I'm sure they have an elevator and you should surely check, as the tour is one of our highlights of our visit to Stockholm.
Restaurant/Bar recommendations include Gondolen Bar - overlooking the harbor, Fotografiska Cafe, and Mathias Dahlgren @ The Grand Hotel (separate entrance from the street). Getting to the first 2 can be difficult as there is major construction near the Gondolen Bar. You can take the hop-on, hop-off boat to the Fotografiska and walk down to the Gondolen Bar. There are 2 elevators to the top and the top 2 floors have an outside viewing area and bar; the floor below has an enclosed restaurant and a bar with an eating area. Of course by the summer of 2017, the construction may be completed in that area.
We did not explore the Stockholm airport but it is 50 minutes outside of town. There is a tram going into Central Stockholm.
Enjoy your trip and happy planning!