The british Island has been invaded by several european peoples, but not at the same time. Each time there were like 500 years in between, so the languages of the invaders had time to spread and develop.
"sick" is a saxon word
(to this very day the dutch word is siek, the local northwest-german word is siech).
"ill" is a skandinavian word, it means "bad" and also "sick". The word came with the Normans to Britain, about 500 years after the Saxons had taken the island.