As another poster noted, Mons Klint is a real pain to get to without a car. We had a lot longer in the city than you do, and ultimately chose not to do it, as beautiful as it looks. Also, we contemplated going over to Malmo, but after reflection, we recognized that we were only thinking of going to "cross another country off" our list, and decided our time would be better spent elsewhere. If those are "must sees" for you, then of course, that's your call.
If I were you, i would try to do two full days in the city itself, and then two day trips. Personally, I think one full day trip should be an early train to Helsingor for not only Kronborg, but also the excellent maritime museum (http://mfs.dk/en/, which I think your kids might really enjoy). If the weather is nice, go to Kronborg first, stop at a market and have a picnic lunch on the grounds, then go to the maritime museum, then train back to Humlebaek for the Louisiana Museum (note that Tuesday through Friday they are open until 10 p.m.).
A second excellent day trip would be to Roskilde (in fact, this is really only like a 2/3 day trip, so you'd still have time for some fun in Copenhagen). It's only about a 20-25 minute train ride from Copenhagen. Get to the cathedral in the morning (a UNESCO World Heritage Site where Danish royalty are buried), then head down to the awesome viking ship musuem (your kids will LOVE this place). Have lunch in Rosklide, then walk around the water and the park in between the cathedral and the museum. Train back to Copenhagen, you'd be there by mid-afternoon.
There is plenty to do in Copenhagen itself the other two full days you have. For the boat tour, do the Netto Badene one - it's the exact same tour as the more expensive ones. Another thing you might consider with the kids is the Exploratorium. When I went (2.5 years ago) it was in a temporary location in Christianshavn, but now it is in Hellerup. I think you'd want to see Rosenborg and its gardens. Also, a good way to burn off some of your kids' energy is to head up the Rundetaarn. Definitely Tivoli too. In Nyhavn, there is a great ice cream/waffle shop right on the canal called Vaffelbageren - grab a treat and hang out there.