Congratulations on your anniversary! The last time I was in Copenhagen, I stayed at the Palace Hotel. Absolutely gorgeous spot and very modern. It was definitely romantic. There is a restaurant there, but I would recommend that you go out and experience other restaurants outside of the hotel (though I do recommend the hotel's breakfast buffet).
One of my most memorable dining experiences was at Restaurant BROR, which is a modern Danish restaurant that was opened by folks who used to work at the famed Noma. How can you go wrong when the guys who started this place used to be employed at Noma? Not only are the dishes here incredible, but they look like little works of edible art. Seriously, you’re going to want to frame them! They basically serve a set menu of appetizers, an entrée, and a dessert. I’m not that into wine, but I went with a few wine lover friends who said the wine list was excellent. Definitely make reservations as soon as you can because it is a very busy place.
I hope you enjoy your anniversary trip!
If you aren't afraid to drop some money, Nimb hotel is a good option. It is very beautiful and romantic, has good restaurants and has the added bonus of being connected to Tivoli which I referenced on your other Chrismat market question. It is also close to the new, hip restaurant and bar scene in Vesterbro . It would be extremely romantic palce to stay at this time of year.
On the high-end/expensive side, Hotel D'Angleterre is a great option. I think this is the best location in Copenhagen, at the end of Strøget , close to Nyhavn and a bunch of other sites in Copenhagen.
Other hotels that have been highly recommended by others are First Hotel Skt. Petri , Admiral Hotel and the Avenue Hotel (the location isn't quite as good as the others, but still close to the main sites).
You already had some food suggestions but if I should add an additional one then it would be Admiral Hotel in Copenhagen which also has an excellent restaurant and the hotel has an excellent view to the opera house and is right by Nyhavn which is both pretty and lively.