Posted: 21/01/2015 at 19:16
Have a look here http://www.visitnorthnorfolk.com/things-to-do and see if you think you'd like the area - there's lots of seaside towns and villages along the North Norfolk coast, and lots to do there and inland too - you can vist Pensthorpe http://www.pensthorpe.com/ one of the homes of Springwatch where there's lots for families to do.
And the beach at East Runton is the most child-friendly one I know, our family's been enjoying it for generations - lovely soft sand at the top, then when the tide goes out there are rock pools with crabs and winkles and lovely firm sand for building sand castles and playing ball games. Then when the tide starts to come in again there's lots of shallow water with a sandy bottom for paddling and splashing in http://www.explorenorfolkuk.co.uk/west-runton.html
And the drive from Sheffield to Norfolk really is quite straightforward - I've done it quite a few times - and it doesn't take too long so you have time to stop every so often and let the little ones stretch their legs and get some fresh air. I suffered from travel-sickness as a child so I know what they're going through