WI: The North Sea Empire survives

  Well that would need a unified succession rather than it being split with a different king in England to that in Denmark - can't remember if Norway managed to gain independence at this point?  Also it would need that monarch to be less brutal that Cnut.  Rather than taxing England heavily to fund the rest of his empire and his military operations and the widespread exporting of slaves from England you would need a monarch and his successors to treat the assorted kingdoms more equally.

  Even then you might see moves towards independence at a later stage.  When the Danish dynasty died out Edward -[the Confessor] was widely welcomed despite the dynasty having been absent for several decades.  Similarly when he supported Harold's claim to the throne the latter had no problems securing the throne.

  However assuming it does manage to stablize for a prolonged period then:

1)  There would definitely be a closer social, linguistic and other similarities to Nordic culture and probably more Danes settled in England.  There was already a significant number, possibly a majority in much of the Yorkshire area.

2)  Possibly although the Swedes are likely to resist this vigorously and it would take a lot of resources and men away from both England and also possibly from areas such southern Denmark which might be vulnerable to attack from the empire.  Also occupying this region could be a long and costly process before the bulk of the population are resigned to foreign rule.

3)  With the Alfredian dynasty withering in exile there's going to be no real excuse for William of Normandy, assuming he still exists and becomes Duke of Normandy to launch an attack.  Also if the kingdom is united then there's no separate Norwegian invasion, or distraction by Tostig so the Normans are going to meet the full English strength possibly also backed by forces from the rest of the empire so is likely to get stomped heavily if he tried such a move.

4)  Agree its difficult to see what happens in France.  There won't be a powerful noble who also rules lands in Britain as a separate monarch to challenge the monarchy in France.  This could help the king centralize power or the lack of a clear threat, which because its core territory is outside France could be seen as a foreign one so it could go either way.  Assuming developments OTL are reasonably the same with N Spain resisting and gradually pushing back the Moors, Italy and Germany both divided then unless the Empire of the North gets involved there isn't a major external power to pose a threat so the French nobles could continue to squabble with any monarch that starts getting too powerful being seen as a threat by his lords and challenged.

5)  Without the chain to Normandy and the reduction of English power there would continue to be greater links with N Germany and the Low Countries.  If an Hanseatic League equivalent is established it would depend on its connections with the Empire.  The latter would be a big possible trading partner but could also, as a powerful state, be a major rival that the League is organised in part to oppose.

6)  Relations between England and Ireland were generally pretty good as they were distant enough from England not to be a great problem once Irish and then Viking raids from Ireland stopped.  It might be that a powerful English state that has secured control of Wales and Scotland would seek to extend power or at least some form of influence over Ireland as well but that's likely to be centuries ahead at least without the Normans.

  Wales and the Scots are likely to be continually at odds with England with raids by them and the English seeking to either punish such or control them directly or indirectly.  I did read once that the primary incentive for the Scots raids on northern England was in part the poverty of much of the country meaning that loot and slaves from the south were very important for the economy and a similar issue could be the case with Wales.  At the same time the English as well as seeking to protect their borders could have some regional nobles who themselves want to gain land, resources or prestige from attacks on the other nations while its quite possible that a strong monarch, in response to Scots raids would at least regain Lothian for England which would deprive Scotland of its richest and most English territory and push back the attacker or could go for a fuller conquest, or at least Scottish recognition of the primary of the English monarchy as an overlord as in previous centuries.  Being smaller and nearer to central and southern England Wales is very likely to be conquered and possibly settled over time although the north might stay largely independent for a fair time, sat several centuries.