A week is fine, especially during the summer. Best daytrip goes to Helsingør, Elsinor. From Copenhagen you can easily reach a number of nice beaches incl. swimming in the now clean harbour. I am a homeexchanger myself. Great way to travel. Also with children, you have so much better space than a hotelroom. Incl. a kitchen, so you can avoid the horrible restaurant prices. When you go to Tivoli, remember to eat before and after.

Plenty of daytrips -- In addition to Helsingor, there's Roskilde for the viking ship museum and cathedral with tombs of monarchs going back many centuries, also Frederiksborg, the castle built mostly in the 1600s that's now the Museum of National History (fortress, stained glass and other amazing decor, gardens, suits of armor and weaponry).

A week is not too much, there is plenty to do and see, even once the main sights are exhausted there is the beach or walking/cycling through forests or seaside villages, you can also take a daytrip to Malmo. Some of the northern suburbs of Copenhagen are worth a trip although not on the tourist trail. Møns Klint makes an interesting excursion.

If I should suggest just one church (children normally get tired of them) it will be vor frelser kirke in Copenhagen, area Christianshavn with the spiral staircase on the outside. Great view when you reach the top.

A week is a great amount of time. In addiiton to the day trips listed above don;t forget that you can easily get to Malmo in Swededn for a day in a really cute town.
Plenty to do - esp Tivoli at night.

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