Travelling to Denmark with family of 5

Billund is quite small. You will be seeing people, so perhaps 3 days is fine, but you will see ALL of Billund twice over in that time, even with the trip to Ribe. So it really depends on how much Lego stuff you are doing. There is a TON of Lego and not much else to see in Billund. Is your family associated with Lego?

Kids have a variety of interests - military history

Consider a visit to Hanstholm fortress museum with Europe's largest WWII fortification.

Beaches and dunes between Blokhus and Løkken are worth visiting at sunny days - you are allowed to drive on beaches.
Rubjerg Knude Fyr is a special lighthouse.

COPENHAGEN

Consider taking the scenic coastal commuter train to Louisiana Museum of Modern Art.
Consider taking the scenic coastal commuter train to Klampenborg and the more workaday
Bakken Tivoli amusement park. You'll walk through a lovely oakcovered park to get there, and by the train station there
is a lovely beach (nude bathing at the far end). Vegetarian buffet at RizRaz.

AALBORG

Went there in 2022. A slightly boring city, in spite of the nice park behind the train station, which is Aalborgs best feature. The vegetarian Indian food of Aalborg was the most expensive we had ever had. As options were extremely limited, we had to have it, which was an unwelcome surprise. Aarhus is probably a much better choice.

SKAGEN/FREDRIKSHAVN

Park Hotel is a good place to stay in the area. I've stayed twice and will again. They have a nice
breakfast and the best line of toiletries (I Love Eco Essentials) of any hotel, and I've been to quite
a few. Vegetarian pasta at La Vida.

Firstly, my thought is that you should probably rent the car in Jutland instead, that will save you a boring drive and and a large bridge toll. And i recommend the train from Ålborg to Copenhagen instead of flying.

For art in Copenhagen, don't miss the Glyptotek. A day trip to Helsingborg is also not a bad idea if you like castles. And if the kids like marine biology, a visit to Den blå planet just outside Copenhagen is probably as close to a must as possible. https://denblaaplanet.dk/en/ You also mention that they like art, any particular favourite eras or styles?

I also agree that 4 nights in Billund is a lot, outside Lego there isn't much to see there. And spending a day to explore Billund might be very disappointing.

Thanks for all of the responses so far. For the time in Billund, I anticipate we will spend a full day with family, another day at Ribe, and then a partial day touring Lego House. Given the kids' ages, I dont think we will do LegoLand, since I've heard it is not that exciting for teens. Since we are coming from a full week in Copenhagen, I also anticipate that everyone will need a bit of rest time just to recharge.

That said, if there is anything else within driving distance that is worth checking out, would love to hear about it.

For your teen who likes military history, consider visiting the Dybbol Banke History Center near the German border. While you are in Billund, you could easily go up to Aarhus where there is a lot of interesting stuff, including the Den Gamle By open air museum and the Moesgaard Museum. The rune stones at Jelling (practically next door to Billung) are also really interesting.

Instead of four night in Billund, you could rearrange those nights a bit. Take the train from Copenhagen to Esbjerg. Spend two nights in Esbjerg and do a day trip to Ribe and visit Esbjerg's art museum. After two nights in Esbjerg, rent a car and continue to Billund and spend two nights there.

If it fits into your itinerary, I like Arhus for the museum with the Graubelle Man. I was there when I was not much more than a teenager (ie, many years ago!) and if your kids like science, they may enjoy this. .

I've been to the aquarium in Hirtshals, and I was not impressed. We went because our friends from Aarhus met us coming off the ferry, and we didn't want to turn around a drive back right away. If I were in your shoes, I would go to Skagen to see the Skaw rather than Hirtshals.
Another vote for Aarhus. In addition to the sights mentioned by others, the art museum is excellent, and DOKK1 is a very cool building with a robotic parking garage. There's a building that looks like the Millennium Falcon (or something from Star Wars, I don't remember) from above. So much to see and do there.
Have a great trip!

Late August I think your teens would love to spend a day at Islands Brygge, swimming, diving and maybe kayaking. And being seen by Danish teens. (Just saying.) It's a really fun place.