I live there OP- great place with young kids- superb modern metro with lifts from street
Here's my tips with young kids
Bakken plus beach- Bakken is the oldest funfair in the world with lots of things for young children , lovely cafes and all set in Hansel and Gretle type woods , 3 minute walk away is the beach at klamenborg -white soft sand plus shade (( 10 minutes out of Copenhagen on train)
Experimentarium at Hellerup (train)
Acquarium
Beach at Amagerstrand (metro) great white sand beach plus activities and Fort
Tivoli- expensive but good and very central
Nice walks along river and lots of parks
Food
Hot dog stands everywhere- best one is next to round tower in centre - amazing too
Cafe Vivaldi chain-- huge portions, you can share with kids they are that big!
Bones- American diner chain quite cheap, lots of all you can eat- good healthy kids menu
Broens (someone mentioned above) lovely open air food market next to docks
Reffens- similar to Broens but bigger with DJs etc (go on boat from harbour)
Grilln burger and Jaggers-- very good and affordable posh end burger chains
MaccyDs - the one at KongensNytorv is suprisingly decent- better than uk