How much French influence is there in English civilization, compared to the Anglo-Saxons?

It's pretty clear that William the Conqueror completely transformed the trajectory of English history, particularly with the English language. But did that extend to making English society look like French society into the present day, but just without speaking French? Also, how much influence of Anglo-Saxon society remains in England today, especially when compared to Anglo-Norman cultural influences?

The one other thing I would like to point out is that the Anglo-Normans were the ones who began the process of conquering Ireland and Wales, almost in parallel to Charlemagne subjugating the Germanic tribes.