Dane here - I live in Copenhagen
Not sure when "mid-September" is, but just as a heads up - Tivoli Gardens closes on Sep 20th. If you're here before then it's WELL worth the visit though. Rumour is it inspired Disneyland
Don't bother with the Little Mermaid - it's tiny! If you do the canal tour (recommended btw) you'll get to see her from the back anyway.
The Round Tower is worth a visit - awesome view from the top.
If you're into museums at all, I highly recommend Torvaldsen's Museum - gorgeous sculptures! Of course, you can see the best ones for free if you visit the Church of Our Lady.
The best bar around is Bar7 on Studiestrĉde 7 - a stonethrow from both the City square (which also has a statue of H.C. Andersen) and the aforementioned church
If you're interested in alternative communes, give Christiania a visit. No photos allowed though! And stay away from Pusher street (yes, that is exactly what you think it is. No hard drugs though).
Try a real
Danish pastry - called wienerbroed or bread from Vienna here - and you'll never go back!
For an awesome meal, try out Copenhagen Street Food on Papirĝen (Paper Island) - sooooooo good!
If you have any questions at all feel free to shoot me a PM. I'm more than happy to help!