Last year I had an extra day in Copenhagen last year before meeting a friend. 16hrs is a good amount and you can do a lot. The problem is your timings. Get the later train. Getting the train onward is dead easy.
Arrival at 6.30pm is great. The train is dead easy into the city and only about half an hour. If you are hand luggage only you can be into the city for 7.30 - 8.00pm.
The meat packing district generally has a bunch of really good restaurants but it gets very busy. The area is pretty trendy and not necessarily the cheapest but if your alternative is Nynhaven it's much of a muchness in terms of price. The Meat Packing District opens late so if you want a night out it's fab. If you want to go for a drink I highly recommend mikkeller which is close to the meat packing district. It's one of the best breweries in the world and has a bunch of great beers - it's pricey but if you go for a couple of 1/3s for tasters it's great. It's a bit of a walk from Nynhaven though and that would be my reservation. My recommendation would be to go straight from the station for dinner BEFORE you check in because of your timings - this would work with eating out in the meat packing area. Getting food much later you may struggle as the Danish tend to eat fairly early. I would book ahead of you can - a table at 8.30 would work.
Then go to your hotel and have a potter around the harbour in the evening. That way you get to see more of the city. Walking around is fine - it's a safe city and having been alone it's fine.
Personally I wouldnt have the little mermaid on the top of my list to do - there's better things but you can do it as part of a walk up to the old barracks and the windmill and that's nice. And you can do that early. As long as you do that rather than just the mermaid (which is a massive disappointment) it's fine. But it's a nice walk and you won't get time to do much else.
The Roundtower is a great little thing to do. Again in the centre. You have to pay to get it but it gives you great views of the city. It's not something that takes that long either. You might be able to squeeze that in at a push but it's one I'd play by ear.
The city doesn't really wake up until 10am. I would suggest doing that then walking back to the station getting a Danish pastry on your way at one of the cafes.
Tivoli is Tivoli. It's great but there's better times of year to go and if you don't like rides, I'd argue there's better things to do. (It's a must at Christmas for me). Wouldn't be on my list for such a short visit.
Honestly it depends on how chilled you are and how much walking you want to do. If you want to really make the most you can but I appreciate not everyone wants to run around like I do on holiday.