Is the reason English is closer to Danish because the Anglo-Saxons spoke something like that or is it because of Vikings conquering parts of England for a while?
Most of the Norse who subsequently invaded in the 700-1000 were either Danes or Norwegians[2], and settled in pretty large numbers in East Anglia, Northumbria and Mercia.
Then the Normans, who were in large part Danes and Norwegians that had gotten bought off with a piece of France came over in 1066.
I don't know how accurate this is, but in Bernard Cornwell's Uhtred series, English and Danish are different enough that they aren't really mutually intelligible.
[1] https://en.wikipedia.org/wiki/Angles#/media/File:Angles_saxo...
Also north east England was under Danish influence for quite a while.