I went in October/November? - can't remember exactly and it was surprisingly warm. We spent a weekend there with 3 kids and they didn't get bored, but they are older than yours.
You can get a tourist pass that gives you free train travel, bus travel and free entrance to the museum, Tivoli and zoo.
Tivoli amusement park - right in the centre of the city, near the station. Good fun, but expensive. The tourist pass gives you free entry to the park, but you then have to pay for the rides.
The museum has a special children's section - dressing up, life-size replica Viking longboat to play in, section of a classroom, ship's bridge, castle-y thing I think, Indian village house. My kids enjoyed this more than anything else we took them to!
The zoo is good too, and big. They've got little pull along trucks that kids sit on and make their parents do the work - you pay a small amount to hire them. There's a special kids' section of the zoo with pony rides and a big play area.
Food - I can't remember the prices but I think they were on the high side. Our hotel provided breakfast and a buffet style evening meal (it was a business type hotel) so we only had to pay extra for lunch.