Right in the heart of the city is Tivoli Gardens, a amusement park said to have inspired Walt Disney to build his own. Next to it is the train station, city hall square and one of the longest pedestrian shopping street in Europe. In terms of scale, it's a fairly walkable city and all the entertainment you need is located right in the heart of the city. There are beautiful parks, and open squares all around the city in close proximity.
Of course there are areas of neighborhood surrounding this island/peninsula of the City, but I think a city can only be alive not because of its ideal location, but what it can offer the people who live and work in the city, as well as the amenities and entertainment that will attract people into the city during the weekends and holidays. I have the benefit of exploring the city during a bank holiday and a work week, and I must say that even with most of the shops closed, there is very little difference between those two days. In fact I didn't even know that I was in the city during a bank holiday, but I was certainly curious why most shops were closed.
I'm on my way to Stockholm right now. Both cities are about 5 hours apart by train and the travel between the two city is really easy and straight forward (just hopped on the train in Copenhagen, do a switch in Malmo and you're on your way to Stockholm) as there's no border control because of the Schengen agreement.
What's next for me in Stockholm? I do not know.
*The picture above is the famous Little Mermaid statue in Copenhagen.
*Two interesting notes, I did not see any gas station inside the city limits and there are no street lights on the side of the streets because they are hung high up in the middle of the street.