The Botanical Gardens, close to Nørreport Station, have always been a favourite of mine. There is so much birdlife and wildlife packed into the park. I have even spotted little turtles basking in the sunshine by the side of the lake. It has the largest selection of plants in Denmark and has recently undergone a major renovation (my photos are from a few years ago). I will definitely be heading here camera in hand once the weather brightens up.
You can stop by one of the bakeries and cafes by Nørreport station for a small treat and sit on one of the benched dotted around the lake to relax.
I am always amazed that, despite its proximity to some busy roads, how quiet and peaceful it is here. It is a great place for a romantic stroll or a place to sit quietly whilst your baby sleeps in the pram (something I did a lot)
The tropical hothouses are also worth a visit but be aware they close fairly early in the afternoons.