The North Sea Empire of Cnut
Imagine that Cnut lives until he is 60 and dies around 1055, he is succeeded by Harthacnut/Cnut III, who is a much more competent ruler and lives until 1081, being succeeded by his own son, also imagine that somehow the If the North Sea Empire survives and remains united in the figure of a single monarch for at least two centuries, how do you believe that the English language and culture would be affected by continued Norse domination?
The North Sea Empire of Cnut
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I think this is the best waysuccessful keeping it together through wit and threat of violence for several generation until the new status quo are accepted.
The Scandinavian influence on English wouldn't that much greater than OTL. The Anglo-Danes were largely assimilated by the 11th century and it's unlikely there will be further waves of Scandinavian settlers. And unlike William the Bastard, Canute wasn't against appointing Englishmen to high offices. English wouldn't be reduced to the language of peasants in this ATL.
Regarding grammar, the biggest difference might be the plural system for English. The Normans didn't introduce -s for plurals in English, but it did become supreme under their rule due to the influence of French.
Verb conjugations (I am, you are, he is, etc.) could become a lot more simplified in comparison to OTL. There would be a need to ease communication between Old English and Old Norse speakers given the confusion these two similar languages would create for bilingual speakers. This had already happened in the Danelaw, but now Danish rule has extended over the whole country. It's possible English could end up like Norwegian with only one verb conjugation for 'to be'. Something like: I are (Jeg er).
It's also possible English would start to influence the North Germanic languages if Anglo-Saxon noblemen and traders begin to move into Scandinavia.