Check the opening times!
Second half of august is good because the danish schools have gone back. Sunday is always quietest. Check if there are any events on like ferarri day or emergency services day because the park fills up for those.
The rides close before the park does so you might want to save miniland for the way out. Or you could go and look at miniland in the evening when it is free to get in. Its usually from half an hour before the rides close.
Ive heard only good things about lego house. They have a free build on i think the first thursday of each month. Ive only seen it advertised on fb.
Places to stay
Legoland village - they have huts, some are themed.
Lalandia across the street has lots of holiday lets.
Hotel legoland.
They are building a new lego castle hotel right by the park for next year.
Nearny towns/villages are Hejnsvig, Grindsted, Vorbasse/fitting, Give
Eating in the park
I would take a packed lunch and then have your evening meal at one of the restaurants. Danes are good at buffets. Its about 200kr pr adult including drinks, cant remember the kids price. Theres a pizza pasta place by the penguins, the castle, the family buffet. Theres a steakhouse in the wild west, captain rogers fishnchips.
Other stuff to do in the area:
Givskud zoo
Drive to Ribe its v pretty olde worlde denmark
Go for a walk on grene sand or randbølhede. Both a few km out of billund.
Aquadome at lalandia, also iceskating soft play bowling minigolf.
The sculpture trail through billund is quite nice. So if you were staying at one of the above places you could walk to legohouse on it.
If you want shopping go to Vejle.