Not to nitpick the question or anything, but the first person to take the title "King of the English" was Æthelstan, who was crowned in 927. There are arguments over who exactly was the "actual" first king of all England, dating back to Alfred the Great, but Alfred and his son Edward the Elder (Æthelstan's father) never ruled the entire country. Æthelstan's successors, including his brother Edmund I and a succession of rulers descended from Edward the Elder and Edmund I, controlled England until the reign of Æþelræd Unræd(1) when Sweyn Forkbeard briefly ruled, and again once Æþelræd regained the throne. Æþelræd's son Edmund Ironside briefly ruled the kingdom until his defeat by Cnut of Denmark, when he was reduced to Wessex, which Cnut took over upon Edmund's death. Cnut's successors ruled England until 1042, when there was a brief "Saxon restoration" led by Æþelræd's son Edward the Confessor and his successor Harold Godwinson; Harold of course was killed at Hastings.

1. This is often translated as "the Unready" but the old English means something more like "ill-advised."

I'm slightly tired and post-pub, so forgive me if I return to this in the morning. For now though:

This is a complex question. Normandy continued to be important to English monarchs for centuries. During the reign of Henry VI, for example, his uncle the Duke of Bedford, was regent of the French territories. The most important of these (as attested by Henry V's will) was Normandy. This was because of the connection to the Conqueror, as well as for practical reasons. Normandy never stopped being the lynch pin of England's claim to French crowns.

However, from William I, the crown was English. William was not a 'Norman king' in the sense that Normandy was a dukedom, and therefore the right to a crown came from England. This was true under William, his sons, and under Henry II and his sons, to whom Anjou held the place that Normandy had meant to William, though perhaps to a slightly different degree. Henry II held or claimed lands that stretched from Ireland to the Pyrenees, and his title, if not his crown, came from Anjou rather than England, just as William's had come from Normandy. There was no gradual move from 'Norman' to 'English' kingship. This was true politically and 'culturally'. French was the language of court for a long time after (please forgive me for not checking the date tonight).

At what point was the king 'English'? In the centuries after 1066 Stephen and John (interestingly both highly unpopular within England) stand out as spending the majority of their reign in England. John lost Normandy in 1204, but he and his descendants didn't give up on France. Perhaps one should consider the reign of Henry VI and the end of the hundred years war as the point at which the English monarchy gave up a claim to the French crown.

Essentially, the monarchs of England continued to think of themselves as both English and French for centuries. This wasn't entirely based upon Normandy or Norman rulership- other French influences intruded also. At what point one would consider a ruler 'English' is difficult. Stephen was certainly a more 'English' ruler than Henry I, yet Henry II (who post-dated Stephen) was Angevin.

Personally (and I may disagree with myself come morning), it took several generations for the monarch of England to be called 'English'. I would also ask what you mean by 'English'. I'm taking it to mean shaped by English politics and culture rather than abroad, but it's important to understand that William I certainly didn't usurp a 'kingdom' that England then reverted to post Norman influence. Anglo-Saxon England had been dominated by Scandanavians, and after William I became an Anglo-Norman/ Anglo-Angevin hybrid. In terms of language and culture shared with his people, the English king wasn't 'English' for centuries, yet I'm not sure that anybody would call him 'French' instead.

Edit: John lost Normandy in 1204 not 1405. I'm going to go to bed now and return to this in the morning. Apologies.