/u/BRIStoneman makes the case here that by the time a unified England was formed, the Angles and Saxons had integrated to the point where they no longer distinguished themselves as such.

Curiously enough, the English themselves seem to have actually stopped making such a distinction by this point, probably because of the arrival of the Vikings. A century of war with the Vikings had led to a strong alliance between Wessex and Mercia which meant that instead of "Angles" and "Saxons" they were increasingly referring to themselves as Anglicarum or Angelcynn, as one common people faced with an external and definitely foreign threat in the Danes. The idea of a single "England" is a concept which stretches at least back to Bede in the 730s, but now we get the idea of a single "English" people, especially compared to the Danes. For example, after King Alfred rebuilds the Roman fortifications at London in 886, he entrusts them to the Ealdorman of Mercia to defend, and receives the oaths of allegiance from "all of the English people who were not under subjugation to the Danes." By the time of his grandson Athelstan's reign, there was a single united kingdom and an idea of a single English people.

Not quite your point, but other languages did refer to the unified English as Saxons, leading to the Welsh "Saesneg" and Gaelic "Sasannach".

While in the English language the country is named for the Angles, that is not true in the languages of the people nearby. For instance England is still known in Irish and Scottish Gaelic as Sasana and an English person is called a Sasanach. And in Welsh an English person is called a Saesneg. All from the word Saxon.

And what happened to the Jutes?

This is a really interesting question and it's been answered on the sub before, but there are a couple of things that I think we can add to the discussion. As /u/GeorgiusFlorentius and /u/ServerOfJustice have pointed out, the term begins with Bede and gradually solidifies until it's standard in the 9th century.

Bede's gens anglorum is a phrase he uses multiple times for a variety of purposes. Often, he's simply referring to the Angles of Northumbria, but Nicholas Brooks (Bede and the English, 1999 Jarrow Lecture) makes an excellent case that he is more often referring to the people of all the Anglo-Saxon kingdoms (as a part of Christendom). Here's a quote from his conclusion:

"What we can, I believe, be quite certain about is that we should not think of ourselves as 'English' today had Bede not written the Ecclesiastical History. His was the powerful expression of a common history, that persuaded those who had been coming to think of themselves all as 'Saxons' that they were instead 'English'. His was the potent message that the 'English' had a clear Christian mission. The fact that the continental English had by 731 ceased to exist as a separate north German gens with their own kingdom, while the continental Saxons still remained independent but obstinately pagan, may have reinforced the appropriateness for his readers of Bede's concept of pagan Saxons but Christian English."

One more reason comes in that 9th century. The greater distinction by then was probably one of Dane and English rather than Angle and Saxon. There was actually a great talk given on this very topic at the Medieval conference in Kalamazoo last weekend. The speaker asked what ethnic identity should we give a 3rd generation Danish immigrant living in East Anglia? His mother might actually be native English and he may also have an English wife. What do we call his children? Are they Danes? English? He brought up a number of entries in the Anglo-Saxon Chronicles of battles and truces between the English and Danish kingdoms and found something really fascinating. When East Anglians (or Northumbrians) break a truce or rise up in rebellion or simply attack one of the English kingdoms, they automatically are referred to as Danes. When they make peace, those same armies and even same leaders are referred to as East Anglians or Northumbrians.

This fits really well, I think, with Brooks' reading of Bede's impact. And I think it also adds some insight to the earlier comments on Irish, Welsh, and Cornish words for the English that call them Saxons. It certainly indicates that English was a self-chosen term.

Lots of scholars have written about this, so if you're interested in the topic, you might want to check out Sarah Foot, Nicholas Brooks, Nicholas Howe, and Patrick Wormald for different takes.

Another, somewhat related question that hopefully somebody can answer if they come across this. How did Saxony come to refer to such a broad area in Germany? There is the Free State of Saxony, which seems to correspond to the borders of the former Kingdom and Lower Saxony and Saxony-Anhalt which together seem to make up Old Saxony. What happened here?

I'd like to ask a follow up:

The Viking/Anglo Saxon divide is very pronounced within the archaeological record (especially with regard to material culture). Were the Angles and Saxons culturally dissimilar enough for it to be visible in the archaeological record?