As there are numerous flights from the UK to Billund, here are some ideas for Billund/Jutland.
Staying in Billund (Lego Hotel, Lalandia, Zleep Hotel etc.) obviously gives easy access to Legoland and the Lego House. Both are great days out - older kids and adults will probably get most out of Lego House, but it's good for younger children as well.
Less than an hour by car gets you to Silkeborg (Aqua, Danish lake district, Tollund Man in the museum), to Vejle (Økolariet) and on to Kolding (Koldinghus Castle, Kolding Storcenter for shopping), or to Aarhus (AROS, Tivoli Friheden park, Mosegaard, The Old Town museum etc.)
90 minutes driving west gets you to the Jutland west coast (Henne Strand, Esbjerg, Blaavand etc) - or east to Randers (Rainforest domes) and Djursland (Ebeltoft, Djurs Sommerland amusement park) Eco homes and a Michelin green star: sustainable living on Denmark’s Djursland peninsula | Denmark holidays | The Guardian Further east is the island of Fyn and Odense (Hans Christian Andersen, Railway museum, Zoo and more).
For quieter places with lots of outdoor activities and cycling, as others have mentioned there are the small islands - Fanø, Rømø off the west coast, or Bornholm way over in the Baltic in the east (fly to Copenhagen, then a train/bus across a bit of Sweden and a ferry to Rønne).
The second half of August is late season (as the Danish kids will be back in school) so accommodation should be reasonable - with the exception of Copenhagen, which is always more expensive than elsewhere. Look at summerhouses as well as hotels, or try a kro (inn) or any of the excellent family hostels.