Sorry you were unable to get your money's worth with the Copenhagen card. When my fiancee and I visited in the summer, we saw enough of the sights to make it well worth the money. I think it was about 165 E worth of admissions (plus the transit pass) for 99 E each. In Amsterdam, I think our total was 143 E of admissions (plus the transit pass) for 93 E apiece. But it did involve fairly precise planning, and luck that we didn't run into any problems (fatigue, traffic, strikes, etc.)
Day 1: cruise ship docked, all the free sights (Changing of the Guard, Marble Church, Kastellet, Little Mermaid, Gefion Fountain, Nyhavn, walked around the Stroget)
Day 2: activated cards for train to Frederiksborg, Baadfarten boat tour, Amalienborg Palace tour, Round Tower
Day 3: Rosenborg Castle, canal boat tour, Christiansborg all-access tours, Thorvaldsen's Museum, Danish War Museum
Day 4: Roskilde Cathedral, Viking Ship Museum, National Museum, Tivoli Gardens
Day 5: we had a morning flight out of CPH, so our passes were still just within 72 hours to use for the Metro ride to the airport
It may seem rushed, but we were more than satisfied with all the time at the sights, except maybe a few more minutes in the National Museum. The Roskilde Cathedral also closed very early for some midday ceremony, so we only had about 90 minutes there.
We took advantage of the CPH public transport online trip planner, and got around the city fairly easily, and minimized waiting at the bus/train stops, although the bus stops are admittedly not well-signed. We did get a bit lost in the shopping streets off the Stroget, but it ended up being a pleasant way to poke around some of the old courtyards and side streets off the main drag.
As for the "museum district," I'm not surprised your bus driver was confused. I think there are more tourist museums near Slotsholmen, which is the opposite direction of Rosenborg. If you said "Norreport," which is the Metro hub near Rosenborg, I don't think you would've had the same misunderstandings. I'm surprised the locals you found were unable to help you with Christiansborg. It would be like asking people in DC to show you Congress, and they couldn't tell you that it was the Capitol building.
Given the physical limitations you are cognizant of, unfortunately I'd guess that most sightseeing passes would not be worth it for you, besides any skip-the-line features, or if they are good for a much longer trip. To get proper value out of them, you really have to triple or quadruple-up on the attractions, I don't think the math would ever add up in your favor if you got only 2 uses per day.
Better luck in the future! Next trip you're off to, let me know and if I've been there before maybe I can give you some pointers.