Go West, Not South
A major Norse empire for it to endure into the 21st century we would need first off to butterfly away the little ice age. Otherwise you loose Greenland and Iceland becomes a cut off minor colony or perhaps independent.
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For the record, learning from what happened later in OTL, a decentralized western grouping would have been far preferrable. Obviously with your POD so long ago matters could have been different, but the way to bet is centralized authority is poison.
For example with Iceland and Greenland is that first in 1262 Norway and then Denmark imposed more restrictive laws and then complete trading monopolies which starved them of profits and initiative, funneling the gravy to royal coffers while telling the people to tend sheep. All hell could break loose and these greedy folk would claim it is heavenly will, while in 1262 the claim was to do the polar opposite.
Case in point was Iceland, where 20% of the population _died_ of famine and to some degree of volcanic poisoning in the post 1783 Laki eruption, and it took nearly a year before the King and his exclusive King's monopoly decided to do anything about it, then spent the money on non famine areas, assumedly of politically connected people. Fish was _exported_ out of the island. You have state control like that and the empire is probably not going to last, and at a distant rule quadriple the chances. Rulers are greedy and are nearly always getting into very expensive petty social squable wars with relatives and colleagues.
Norwegian rule of Greenland did not bode well in the 14th/15th centuries extinction, either. It was a European disease back then, government monopolies, and very few ever gave it up willingly while at profit. A soveriegn gave up those trading rights after the poor population was bled dry, generations afterwards, if at all.
So maybe with some more free enterprise, butterflies are not needed. There was more to it, for example that trade items of Narwhale tusks became not so much in demand (supposed to be related to unicorns, therefore good for alchemy and medicines) or that little matters like the Black Death knocked on most towns' gates. Will you have a Black Death? Certainly plagues, but a strong economy deals with upsets like that. But the isolation could find an outlet with American colonies, sucessful American colonies, if in your TL.
All new Northern areas would be sidelines till that, as most money was otherwise in the perpetual draining military roles down south, fine in warm years with an over abundance of fresh meat young buck warrior in the population of the Viks (harbors) but not so great in down years.
The focus was then on economic freebooting, trade, and hiring out for mercenary pay. Viking trade never took off completely, based enough on muscle and force a bit like Vlad or Ghenghis did, which was an improvement for trade of violent anarchy but requiring contineous inputs and forces.
Your timeline might have to focus more on technology. Besides Clinker Ships (the overlapping boards which provided so much strength and which were used til the late middle ages as standard issue in northern areas), no other technologies come to mind. Any one want to enlighten us on more? Crossbows were not new, but a step towards providing an edge could help a lot towards a modern empire. The guerilla warfare by water will not cut it beyond 999. Otherwise it would be more like the typical conquest routes for the monarchy, something which may pay for some sometimes but is fleeting overall. Eventually some kind of improvement method must be developed, I guess, for a real empire or cohesive confederation to take hold.
If you do have Vikings go south, try the Azores or Madiera Islands, as no one on there (uninhabited til 14th-15th centuries) and capable of supporting as many people as Iceland, Faroes, and Greenland put together, plus only several hundred miles off the coast of Spain. The El Cid types in the Galacia region would be a natural putting off place, and have the islands go to a type of Sicily (Norman rule for many years, plus ruling class for more after that with Vespers, til hacked one last time by the locals in 1730's I recall). Play off both both sides of the Islamic/Christian groups. Empires tend to last when they have a power center, more isolated from other powers the better, which rules out anywhere in Southern Europe.