That does seem to miss the fact that by then there was not a 'Viking king of Northumbria'. The Danelaw did not fully extend in to Northumbria.

My recall of history is that Northumbria had more or less split back in to Deira and Bernicia. With the Vikings having control south of the Tees, but north of the Tyne it was still an Kingdom of the Angles. Albeit possibly a client Kingdom, that seems a bit uncertain.

https://en.wikipedia.org/wiki/Northumbria#Northumbria_and_No...

https://en.wikipedia.org/wiki/Bernicia#Rump_of_Northumbria

https://en.wikipedia.org/wiki/Rulers_of_Bamburgh

So that portion only really became part of a unified England per-se after the Norman conquest, and the "Harrying of the North."

Certainly the English spoken in Northumberland to this day, is apparently heavily influenced by the Old Anglic language.