German and English have almost 2000 years of separation. More than Italian and French and Spanish.
But yes Old Norman (and Old Norse) modified the language extensively.
And interestingly, there's a significant body of research showing mutual intelligibility between Old English and Old Norse, which are less similar than than Old English and Old High German.
Old English texts have been found in (modern day) central and southern Germany (e.g.: Fulda), which implies that they were reading them there. This would be from the 7th to 10th century timeframe, IIRC.
"Saxon" is probably the word that the could have settled on. But that's not really an ancestor of modern high German, more the low German speaking dialects in the north west (Niedersächsisch).
I don't think this is being pedantic, frankly. And it's horrible to say English "doesn't really exist"; English is a beautiful language with a long history as old as (in fact in some ways older than) German. (FWIW I'm a native English speaker but my father's family is German, so while I'm not fluent in German I do have education and personal exposure to both languages. I've also studied a bit of Anglo-Saxon. Enough to see just how different it was from old Frankish spoken by the early "Germans" in the same time period)
Also keep in mind the topic of this discussion, which is a humorous diatribe against the modern German language (written in beautiful, flowing English by one of its best writers to have ever lived). A native German speaker could quite easily take offense at it. I happen to have struggled through obtaining German fluency in my college years and while I wouldn't agree with everything that Twain wrote, I find his piece hilarious.
[1]: "German" being an originally Roman Latin term for a group of related peoples speaking, what amounts to closely related dialects rather than separate languages. The definition of "language" is also really indistinct. It really comes down to mutual intelligibility than anything else, see Norwegian, Danish, Swedish (separate "languages" but mutually intelligible). You're certainly aware that the modern Germans call themselves and their language "Deutsch" -- an old word for "people" and that was the case in English in the past, see the "Pennsylvania Dutch" who most certainly don't come from the Netherlands or speak modern Dutch. Old English had liberal use of the term "þéod" (people) which is cognate to "Deutsch"/"Dutch". The country of "Germany" has only really existed since 1871.
English started as old German. Then vowels shifted. Then the Normans arrived and added a lot of French words on top. When German grammar proved too hard for them that was simplified too.
On top of all that is a bunch of Latin and Greek added when England lead the scientific and industrial revolutions (in part because English couldn't logically supply words for new things). These kept their own rules grammatically. Or they didn't, pretty much randomly.
None of these jobs were ever "finished", so nouns in English don't have genders (like they would in German) but actually some do still (mainly pronouns) and some words right next to each other come from different languages (Cow is German, Beef is French hence they sound nothing alike).
Since England has not been conqoured or undergone unification since before all this started (except for the Norman conquests which were part of this) nothing has every been standardised. This is why spelling English words is all but impossible (German words are spelt almost exactly as they sound, English words almost never as they sound, you just have to memorise them, ALL of them).
Compare this to German or French with logical spelling, strict, consistency in grammar and single root languages.
English really is awful and quite an unusual language.