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Difficulty: Medium
First of all, you have to win the invasion. Get a strong ally but dont make an alliance with the King of France, because he can not join your war. After your victory over England you have to defend against the Norway Invasion of England. After you won both wars there will be the hardest part of the achievement. Moving the Realm Capital within the de jure territory of Kingdom of England enables the Embrace English Culture decision. You can do this when you select a city and click on "move capital here". Now revoke all the duchies of non-English Vassals and transfer non-English counts to English dukes if you have at least one in the realm. You can use one of your sons, but he has to be English (your firstborn does not work because he is Normen). You can call your allies to help you in your wars. In my run there were two duchy I had to deal with. This achievement checks direct vassals only. So you can grant all the small county Vassals to your one English Duchy son. You can see all your direct Vassals in your realm menu.

Credit to Bayek von Kuwaq on Steam for his guide.