Well, the Danes weren't actually "expelled". Danelaw controlled much of the North and the Danes inhabited the North of England largely what was Northumbria. The area was brought into the fold of England under Eadred roughly around 954. But the Danes were still present. They had just settled by that point.

Because the Danes began renewed invasions, Aethelred the Unready ordered the slaughter of all Danish males living in England. Fearing they would be a sort of third column. It's difficult to say to what extent this was carried out but it seems to have led to some significant loss of life. This was used as a pretext for invasion by the king of Denmark Sweyn Forkbeard, who conquered England and became king in 1013 but then died. His son Cnut, after briefly being driven out, became king of both England and Denmark in 1016. He ruled England (and Denmark and parts of Sweden) until 1035 when he died of natural causes. He married Aethelred's queen, Emma of Normandy (mother of Edward the Confessor and aunt of William the Conqueror). Then his two sons succeeded him and died in quick order. So that Aethelred's son, Edward, was allowed to return.

I know that wasn't your question but I think it's important to note that the Danish influence on England didn't go away after the late 800s. They remained in large numbers in some parts of the country. And there were four Danish kings of England. The Danes had an impact on English culture and language. And law, for instance the jury was adopted from Danelaw traditions. Considering the victory of Cnut and then the Normans (Norse descendants) one could argue that the Danes actually won in the end. But that's a bit too speculative for here.

Second, the influence of Anglo-Saxon culture is difficult to answer because it's so broad. But Anglo-Saxon England was one of the most well run countries in Western Europe. It was fairly well organized, it had taxes (which was not efficiently collected through out much of Europe), it had record keeping. And the Normans kept most of this Anglo-Saxon organization. Although they replaced many of the people at the top, of course. So, that much of what we think of as English when it comes to local government is fairly Anglo-Saxon. For example, the English had divided local territories into shires which were answerable to a shire reeve (sheriff). Each shire had a shire court, which helped administer the laws. These are all concepts we are pretty well familiar with and they were Anglo-Saxon bodies.

The Anglo-Saxons were also fond of keeping extensive written law codes and recording land transactions. They weren't alone in this in Europe but they were fairly well advanced. This was also enormously helpful to the Normans.

Although the Normans (and the Danes, to a lesser extent) changed the language very much so that modern English and old English are drastically different, we still speak a form of English not a form of Norman French and certainly not a form of old Norse. The Anglo-Saxon Germanic root is still the base of what would become our language. So, that is a fairly significant legacy.

Celtic/British, Anglo-Saxon, Danish, and Norman literary and cultural institutions have become so entwined that, for me, it's difficult to really track inspirations when it comes to literature and broader culture. Although hopefully someone more equipped can speak to that. To give an example of why it's difficult, look at the legend of King Arthur. A Celtic warrior who allegedly fought the Anglo-Saxons but was reintroduced in popular tale via France thanks to the Normans and Angevins. So that you end up with a work like Sir Gawain and the Green Knight, which uses alliterative verse, popular in Anglo-Saxon poetry and a sense that Sir Gawain is very English rather than French styled. There are hints of the world of fairy popular in native stories (both old English and Celtic). But it's laden with the chivalric themes introduced by the French. It's difficult to pick apart cultural influences once you start digging down.

From a legal and linguistic perspective the Anglo-Saxons did have a lasting impact. And my focusing on that isn't precluding other ways they impacted society.