It seems to me the English translation is off. The sentence should be in the present perfect which denotes actions that started in the past and just ended in the present. Therefore it should be: "I have just recovered from a severe illness." Please advise.
No, it's fine. It's a better sentence for a casual, passing explanation, because, as you say, it places the action in the near past rather than the present. Native English speaker from New England, here.