The Copenhagen Card which you can buy for your stay worked well for us when we visited last December.
It includes access to the major attractions and (excellent) city and surrounding area transport if I recall correctly. Well worth a google given the sites you already mention. The Tourist Info in the city was ‘ok’ but no more helpful than the Visit Copenhagen site.
We enjoyed a couple of boat trips and many sites including Tivoli Gardens (during the Christmas Carnival). We were not overwhelmed by the Little Mermaid.