Go back a thousand years before that. What is "English" is defined largely by the boundaries of Roman Britannia. Scotland was beyond Hadrian's Wall. The Romans canceled their plans to invade Hibernia. And Roman control over (particularly northern) Wales was always tenuous.
And inside Roman Britannia, there was a transformation in terms of genetic history, aristocracy, and culture. The Romans sent in soldiers from across the Empire. And after they had served their 25 years in the legions, those soldiers often settled in Britain with their often-native wives. So the people who populated the major cities like Londinium or Eboracum or Mamucium were half-Briton and half-Dacian or half-Latin or half-Italian or half-Spaniard. They adopted Roman culture--as Tacitus remarked, "All this in their ignorance they called civilization, when it was but a part of their servitude."
And after the Romans, there was the Anglo-Saxon invasion. And after that, there were the Danish invasions. Those peoples didn't eradicate the island's inhabitants; they needed them for labor. They intermarried with them. So by 1000 AD, your average Englishman was a mix of some bits Briton, some bits Germanic, perhaps some Danish, and some bits from elsewhere in the old Roman Empire.
And even before the Romans, you can go back and find evidence of Neolithic "invasions" of Britain from the continent. Over the centuries, that island has seen wave after wave of "invasion" and resettlement.
The remarkable thing about it, however, is the continuity. The Roman roads often followed roads that had existed from the Neolithic Age. Watling Street was a thoroughfare hundreds or thousands of years before Caesar stepped foot in Kent. And even today the lines of farms still track a lot of the lines laid out by Neolithic farmers and adopted by the Romans, the Anglo-Saxons, and then the Normans. Counties and districts are often centered on Iron Age hill forts. And many aspects of English culture would be familiar to a pre-Roman Briton (eels, anyone?). There was genetic and cultural mixing, but the preexisting people and culture was never wiped out. And perhaps that's the most remarkable thing about the people we now regard as "English"--despite all of the "invasions" by various peoples, the amount of genetic and cultural continuity is really quite remarkable.