'Anglo-Saxon culture' is a fuzzy term that's hard to talk about as being just one thing. England before the Norman conquest wasn't culturally homogeneous. Different regions spoke different dialects of Old English, and the peasantry lived in local communities whose traditions and histories were varied. There many 'Anglo-Saxon' Englands, many 'small worlds' that could get along with each other, but which were only very loosely tied together into anything we might today recognize as a nation or culture. Think about how many different accents and cultural quirks there are in modern England today, and then imagine much less unity there was 1000 years ago when there was no television, no radio, no public education, no printing press, and no strong central government to tie things together. Which of these small valleys had the 'real' or most genuinely 'Anglo-Saxon' culture? This label, like all historical generalizations, is very vague, and refers to many diverse (rather than one unified) things.

When the Normans arrived, their approaches to local cultures varied. In many cases, they replaced the Anglo-Saxon elites with people tied to William's court. These new Norman rulers brought new cultural practices with them. They hunted differently, ate different diets, and--most famously--built castles across the landscape to mark their control. Yet they also worked very hard to fit in with the local population. It's easier to rule people who recognize your right to rule them, so Norman elites in, for example, Yorkshire went out of their way to express power in the visual language of their Anglo-Saxon predecessors. Anglo-Norman burial effigies in the North from the late eleventh century look very much like late Anglo-Saxon burials, because the new elite were trying to blend in with the existing systems of power.

Across England, accommodation between the new elites and the people they ruled was the norm. There were never enough Normans to stamp out the Anglo-Saxon population. Instead, the new elites put themselves at the head of English society, and conquered and conquerors shaped each other. Remember, also, that there were many other factors that drove cultural change in this period. England's economy was changing rapidly in response to growing urbanization and trade before the Norman conquest, and both Anglo-Saxons and Normans were swept up in these changes, and their cultures transformed. If the Normans had never conquered England, 'Anglo-Saxon' cultures and societies would still have been hard to recognize by the end of the middle ages from how they looked in 1065.

The language we speak today evolved out of Old English. The Middle English spoken in the 14th century in the west midlands is the bastard child of English and Norman French; some linguists have called it a creol. Other dialects of Middle English (like those in which Gawain and the Green Knight or Piers Ploughman were written) have more Old English than French in them -- but they slowly died out because the West Midlands' dialect proved more successful and, when the printing press arrived, became the language of print media. Old English never stopped being used, per se--rather, it evolved, and we're still speaking one of the tongues it turned into (I'm not a linguist, but I believe the last text written in something linguists would identify as Old English was from the 12th century).

As far as calling England 'England' instead of 'Anglo-Saxon,' this actually happened during the late Anglo-Saxon period. England is Angle-land, the land of the Angles, ie of the Anglo-Saxons.

So, when did Anglo-Saxon culture die? This question is kind of like the paradox of the Ship of Theseus. Theseus had a ship, described by Plutarch 2000 years ago, that had been repaired so many times that every board in its hull had been replaced at least once. Plutarch asked, if all the boards have been replaced, is it still the same ship? And if it is the same ship, what would we call a ship that we built from the worn-out pieces of the original if we were to gather them all up again and build a second ship? Which would be the real ship of Theseus? Anglo-Saxon cultural change is like this. The culture was never stamped out, but it changed, and eventually those changes ended up where modern England is today. Along the way, it picked up a lot of new pieces -- French language from the Normans, sugar from the West Indies, curry from Pakistan, etc, and things like Old English became unrecognizable as they evolved. Does that mean that English culture today is something fundamentally different from 'Anglo-Saxon' culture? Or is it still 'Anglo-Saxon', just with the old, worn-out planks replaced along its journey into the present?

Of course, how we answer the question above often comes down to politics. You may have noticed that white nationalists tend to care more about England's 'Anglo-Saxon'-ness, because it helps them argue that England's people ought to be white, and its traditions Germanic; while those who favor an open society and ties with Europe will point to the Anglo-Saxon era as a time of migration and cultural change, and say that ongoing migration to England is a continuation of that legacy--ie, the fact that curry is now a staple of the English diet is proof that the 'Anglo-Saxon' culture of migration is still alive and well. Which is more historically 'true'? There is no scientific answer to this question, and whether or not you believe Anglo-Saxon culture is alive or dead depends on what you think was 'Anglo-Saxon' England's most defining cultural practice. Which cultural practices from the early middle ages made 'Anglo-Saxon' England 'Anglo-Saxon', and how much do those practices have to change before you think they're dead?

To add to the question, to what extent did the Anglo-Saxon culture and language actually die? How much of the emergent English culture was a fusion of Anglo-Saxon and Norman cultures?