How did the Vikings come to rule half of England?

By conquering it, then settling and signing a treaty. In 865 a coalition of mostly Danish warriors, the Great Heathen Army, invaded England. This was not a raid: it was much larger and aimed to conquer the four kingdoms of East Anglia, Northumbria, Mercia and Wessex. Within ten years the Danes controlled East Anglia, Northumbria and Mercia, leaving only Wessex resisting.

Wessex was held by Alfred, a king whose weak army had at first been forced to pay tribute to keep the peace. He went into hiding, then returned in early 878 with an army and attacked the Danish leader, Guthrum, at Edington. As a term of surrender, Alfred demanded that Guthrum be baptised, and he acted as his godfather.

A later treaty defined the boundaries of their kingdoms and let the Danes govern themselves in exchange for loyalty to England. That is the Danelaw: the part of England, under Anglo-Saxon rule, where Danish laws applied. It approximately covered Yorkshire, the central and eastern Midlands and the East of England.

But 'half of England' is a tidy label for a messy reality. The word Danelaw was not used for a geographic area until the 11th century, and archaeology does not show the area to be a real boundary of people or trade. The Danes came first as an army, and then, in the Anglo-Saxon Chronicle's words, they 'proceeded to plough and support themselves'. Much of it was won back by Edward the Elder and his sister Æthelflæd in the 910s.