The legal end of the Norman Dynasty was when William the Conquerer's Son's Daughter's Son (Henry II) took the throne, as the male line determines the dynasty.

Since this was a legal passing of the throne, the Norman ruling class didn't just disappear. If you are wondering about the end of De Facto Norman-identifying rule, I can't really help you.

Ethnicity is a difficult thing to define at the best of times. In this case 'Norman' isnt even an ethnicity. You could argue that theyre still in power now, they ended when Henry II ascended the throne or when they started speaking English more than French, 200-300 years after 1066. Its entirely subjective.

Even the various dynasties of England (Norman/Plantagenet) are a more modern category retrofitted on people. Its hard to know if William the Bastard, once crowned King of England, still thought of himself as 'Norman' or 'English'. Henry II may be quite surprised to know he was the first 'non Norman' king as the concept may have been meaningless to him. Especially when nationalism was an unknown or rarely thought of concept then these terms would be of little relevance to the people living in those times.