There's plenty to see and do around CPH. The Marriott has a good, central location. You can walk to Tivoli and City Hall (Rådhuspladsen) in 10 minutes. You should go to Nyhavn (new harbor) and see the buildings there, perhaps take a canal boat tour but make sure to get on a covered boat at this time of year. The canal boat tour will show you lots of other architecture around. The Marriott itself is not far from a building called The Black Diamond (Sort Diamant) that is the State Library. A very interesting building. You can walk into town and walk down the Strøget (shopping street) - Nyhavn is close to the end of this area near Kongens Nytorv. It's been a while for me but a walk up the Round Tower (Rundtårn) is also interesting and you get a nice view of the area. Normally in the spring and summer I would suggest Tivoli but right now they are closed so going there won't be possible.
It is quite easy to take the train up to the Lousiana Museum and that has some great art. You could also go to Helsingborg up north and see Kronborg Castle (aka Hamlet's castle). I think it takes about an hour by train from Copenhagen.
If you're interested in modern Danish architecture you can pop by the Radisson SAS there near Tivoli and the main train station.
And gadgets/Danish design can be found lots of places. :-) You'll see it when walking down the Strøget.
If you have a chance, try a hotdog at the Andersen Bakery. It is accessible from the outside of Tivoli, just across from the train station.
For kitchen gadget you want to visit HW Larsen in the old meat packing district (Kødbyen)
The hotel candirect you there - it is a 15min walk from the Mariott.
In Kødbyen you will also find some decent restaurants and bars.