Hi @Luca,
I'm sorry for missing your post a couple day as ago, ref. above [, too!].
The suggestions and recommendations from @frankhkii and @Gamecock-YT are actually very good.
For a weekend stay this coming weekend I would suggest you to simply hedge the weather bringing practical foootwear and a raincoat for you both, so you can give the weather a finger, if needed to, in stead of getting more or less grounded because of it. The weather for Copenhagen at www.dmi.dk looks fairly good for the weekend. But the Danish weather gods have been quite moody in general this year.
If the weather gods are with you, a visit in Nyhavn is [almost] statutory, I would say. Personally, I have also developed over time a soft spot for the area around Skt. Annæ Plads and down to the waterfront, including the area around the outer end to the water of Nyhavn, after Hotel 71 Nyhavn, direction Skuespilhuset, ... It like an oasis to me, when the weather is good!
Rosenborg is also a great place to visit - fantastic garden - so relaxing to stay there!
If you want to hang out a bit for your self - pretty much in private with something to drink [soft / not soft], I would recommend a waterhole at Kultorvet or at Gråbrødre Torv! - The food at the restaurants at Gråbrødre Torv appear to be fairly good, too.
Enjoy!