Oh, you are so lucky… I wish I were going home soon…
And the language will probably not be a problem at all. The vast majority of danes speak excellent English, especially in Copenhagen. You will probably hardly notice it except for the accent, which is very mellow.
I will echo the calls for Tivoli during the evening, where you should sample both Gløgg and Æbleskiver. The former is a hot drink similar to mulled, spice wine but even stronger. IIRC, they have varying levels of “strength” you can choose from. The latter are small, round pancake-like doughballs which you dip into jam and powdered sugar. Heaven!
On a related vein, the christmas beer brewed by Tubor is a national institution*. it’s called “Julebryg”, and has a cute little label with snow on a blue background. Its a dark ale with some spices I think.
If you want to go to a quintessentially danish restaurant, I’d recomment Det Lille Apotek-the little apothecary (http://www.det-lille-apotek.dk/) which is the oldest restaurant in Copen, and marvelously cosy.
However, the amount of amazing ethnic cuisine is stunning, and in my experience it’s pretty hard to have a bad meal. A great sushi place is on Herluf Trolle’s Gade, right off Kongens Nytorv. I can’t for the life of me remember the name, but they have a great grilled lunch-menu which is relatively cheap.
I’d highly recommend Rosenborg, since it’s the perfect little fairytale castle, with a permanent display of the crown jewels which is very pretty.
Also, a stroll along the pedestians-only central shopping street colloquially (and now semi-formally) called Strøget is great. Start at Rådhuspladsen (the town square, near Tivoli and the central trainstation) and work your way to Kongen’s Nytorv. If you like design, stop at Amagertorvet, colloquially known as Storkespringvandet or stork’s fountain due to the fountain in the middle and check out Georg Jensen for beautiful silverwear and jewellery.
The Danish National Museum is excellent and informative, though I’m not sure how thoroughly the exibits are signposted in English. They probably are, but I just can’t conjure up a mental image.
If you want to check out the new Operahouse on the other side of the water, don’t shell out for a tourist-boat (though those can be fun for the rides along the canals), but take a “harbour-bus” instead. they leave from the tip of Nyhavn, and make a circle of about four stops around the harbour, also passing the stunning new public library, called the “black diamond”.
Someone upthread mentioned Christiania, which is very fun, but since the police cracked down on Pusher Street etc, the place in general has become a little quieter, and perhaps just a tad less tourist-friendly. It’s still very much worth the trip, but don’t expect a sixties-ish love-in anymore.
The Little Mermaid is a classic stop, but to be honest, don’t expect that much. She’s considerably smaller than one thinks, and there’s a distinctly anticlimatic feel to the area. There is, however, a modern version of the statue about half-a mile north which is interesting. It’s situated in the area called Langelinie.
If you prefer to take a load off your feet for the night, the London Toast Theatre http://www.londontoast.dk/ is a cabaret that plays in Tivoli, which is an absolute riot. They’re an english company, and the full show is in english, but with a perennial danish character and lots of jokes/jabs at Denmark. It’s basically a classic british pantomime complete with cross-dressing jokes and audience participation, but the wit is of the highest/lowest calibre and you’ll leave with your sides aching. I’d recommend booking ahead as it’s very popular and sells out relatively fast.
The Tourist Ministry has a website http://www.visitcopenhagen.com/ which operates under the name Wonderful Copenhagen, and publishes a booklet ever month with interesting tidbits etc… Look out for it in cafes etc.
I really hope you have an amazing time! Copenhagen’s a wonderful city, you’ll see.
*They actually have a special day where they “release” it ever year, called J-dag.