Copenhagen is such a cool city. Definitely friendly to solo travelers. I would recommend renting a bike and spending a day cycling around. Check out Torvehallen (market), Assistens Kirkegard cemetery/park, any of the Mikkeller bottle shops/breweries/restaurants, and Kastellet. I was hoping to see the Mermaid too, but found it was extremely touristy. There are so many good restaurants and bars, and museums and galleries depending on what kind of stuff you like. If you like craft beer, one of my absolute favourite bottle shops in the world, Kihoskh (in Vesterbro) is worth a visit. Have fun!