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Ah yeah, definitely go fixed in that case! I don't know the West Norfolk coast as well as the other parts, but for a really easy route just following the A149 around the coast is a pretty good option, as it's not really an A road at all. You can't really go too wrong with any of the smaller roads either, mind. Enjoy it!
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You could ride up to Wells and follow the top section of this route (in reverse) down to the broads, it's a lovely route and includes a ferry crossing near Reedham (costs £1)
It's pretty flat in terms of total elevation (highest point in Norfolk being Beacon Hill a mighty 105m above sea level) but as with most roads in the UK, there are deceptive undulations in a lot of places, and you might stumble across a short sharp climb or two along the coast.
I've ridden fixed from Norwich out to the coast too, and it's perfectly doable, but I'd say it's much faster/more enjoyable with gears. There's often a fair bit of wind in Norfolk, and the road surfaces aren't the best in places, so you can often find yourself grinding away for long periods.
Also, King's Lynn is a good 50 miles away from the Broads 'proper', so it'd be a reasonably big day to travel there and back fixed. Unless you're travelling between them in a car?
If you're just chilling out and want to do a couple of short/mid length rides, I'd go with the fixed if you're enjoying it. But if you want to go further afield, then maybe take gears.