So firstly I would caution that it doesn't really make sense to think of these groups as 'factions' in the sense of them necessarily having a united political purpose. Rather they are fairly broadly defined terms for groups of people who tended to share a common language and culture, and would not always be 'united' even within these groups, which might be part of the confusion.

That said, I will try and clear things up a bit for you.

Before the arrival of the Romans, the British Isles were inhabited by various groups of Celtic peoples.

Broadly speaking, based on the languages they spoke, those living in what is now England, Wales and southern Scotland have been referred to as 'Britons', those in mainland Scotland as 'Picts', and those in Ireland and the Scottish Islands as 'Gaelic'.

After the Romans left Britain in the 5th century AD, various people who spoke Germanic languages began to invade and migrate towards Britain. These people are often referred to as the 'Saxons', or the 'Anglo-Saxons', but included many different groups from what is now northern Germany and Denmark.

By the 8th century, Anglo-Saxon kingdoms had come to dominate most of England, while what is now Wales and Cornwall remained dominated by Britons. At the same time, in what is now Scotland, Gaelic speaking peoples from Ireland and Western Scotland had migrated further east into the lands once dominated by the Picts, and by the year 900, the Pictish and Gaelic kingdoms merged to form the Kingdom of Alba, which would ultimately become Scotland.

The Norse invasion of Britain began in the late 8th century, and they would dominate much of Northern England, and occasionally, such as with King Cnut, make claims to kingship over all of England, until the last Norse invasion by Harald Hardrada was defeated in 1066.

In terms of terminology here, 'Norse' and 'Viking' can basically be thought of as interchangeable terms for 'inhabitants of Scandinavia', although the word 'Viking' often carries connotations of referring specifically to raiders or invaders, in a way that Norse does not. 'Danes', refers to a subset of Norse: those from Denmark. These terms were not always used in entirely consistent ways in this period however. The 'Great Heathen Army' which invaded much of England in 865 is often referred to as composed of 'Danes', since the majority of Vikings involved in it were from Denmark, but it contained Norsemen from other parts of Scandinavia as well.

England was then conquered by the Normans in 1066, and the 'English' language and culture which crystallised in the centuries following this contains influence from Norse, Germanic, Celtic and French. So the only possible answer to 'who rules England currently' is 'none of these groups, but also sort of all of them', I think.