I'm the husband and father of the two kids we bought along for this trip; writing this. Copenhagen is such a lovely city that is extremely family-oriented and great public transport so that it is very easy to get around and have plenty to do. I've been recommending Copenhagen for everyone!
As for the apartment itself - it's a brilliant and comfortable place and we've enjoyed staying and relaxing in the place. However if you don't fancy walking up multiple floors then it's not for you but for us, we don't mind it at all and the apartment is worth the climb!
The owner have been a great host in providing a travel cot for on request and there's a high chair and toys in the place which my little one used a lot with the "horsey" soft toy a particular favourite. :-D