Well, immediately around King's Lynn isn't terribly inspiring, but you won't be far from the beaches of North Norfolk. It's what, 20-30 mins up the coast to Hunstanton from King's Lynn. Good beach is Brancaster which is just a bit further over.
Also, before Hunstanton you go past Sandringham which is worth popping in to. You only get to see a few rooms of the house but the gardens are good.
There's also Norfolk Lavender at Heacham on the way to Hunstanton. Possibly not too interesting for the children though.
Holkham Hall, further round the coast (50 mins) is good for a day out. Can hire bikes and explore the grounds. And of course, the beach there is famously enormous and beautiful.
Oxburgh Hall is a lovely National Trust property, about 30 mins from Kings Lynn. Has a good priest hole!
A bit more of a trek (50 mins), but the children will love it, is the Dinosaur Adventure Park at Lenwade out towards Norwich. Dinosaurs in their natural habitat! Fab adventure playgrounds! Loads of space to run around. Favourite day trip for DD and all my nephews and nieces when they visit.
Also, another excellent day out, is High Lodge in Thetford Forest (approx 45 mins). Another top favourite with all the kids I know. Miles and miles of safe cycling on sandy woodland tracks. Bike hire. Cafe. Loads of open space. And the best, and most extensive set of adventure play equipment I have yet come across, all spread throughout the forest. There's also a high ropes site (Go Ape) although only suitable for older children and you have to pre-book. You can even go Segway-ing (again need to pre-book). Highly recommend it.