At that time, the English language was predominantly Anglo-Saxon and remained so for about the next 500 years or more.
A contentious statement to put it mildly.
I wasn't aware that Anglo-Saxon was a language; the language is normally called Old English. It was a Germanic language.
The influence of Danish was considerable. For example the vowel in put and putt is pronounced the same North of a line that more or less coincides exactly with the Danegeld. Had the language of 'Gawain' triumphed over the language of the 'Canterbury Tales' then modern English would have many more Danish influences.
then the Italians (Romans), Germans (Saxons), French (Normans), Swedes (Vikings), and Norwegians (Norsemen) could make the same claim and, frankly, would have more right to do so.
There are scarcely any Latin words that came into English through the Romans; they almost all entered through French or directly through Latin through the influence of the monasteries or universities.
The Vikings were Norsemen, not Swedes (and the languages are similar anyway). The Danes settled and thus their influence was greater but we are talking about very closely related languages anyway. And Germanic and Scandinavian languages are sibling sub-families anyway.
The Normans were originally Vikings, but by the time they invaded they were Norman French speakers. They of course had little effect on the English of the North of England, and none whatsoever on the English of Scotland, which explains why Scots is a dialect of English with many less words of French provenance.
The fusion of Old English with Norman French, which produced the Middle English of the South East, and later Modern English, was more or less complete by the 13th century - considerably less than the 500 years you mention.