There is a Hard Rock Cafe just outside the entrance to Tivoli Gardens. The Hard Rock Cafe is also very close to the main train station. Ask to sit in a back room overlooking Tivoli. This won't be a panoramic view but you should see a bit of Tivoli. Remember, Tivoli isn't very big like an amusement park in America but for its time it was and remains a legendary place.
If you go to the Hard Rock Cafe you will hear English spoken by the Danes and you will surely think you are back in the USA, and the menus are in English for everyone, not tourists.
Also, there is a McDonald's in the main train station. I add this not to be funny but rather to help you if you have children. Copenhagen is a truly wonderful city.
Hope that this helps,