The modern sense of sick dates from Anglo-Saxon times, and by Early Modern English it had developed a number of other senses, applying to objects and concepts as well as people.
An appearance could be sick, if it was "pale" or "wan", as when Romeo describes the moon's livery as "sick and green" (Romeo and Juliet, II.ii.8).
The air could be sick, meaning "contaminated" or "infected", as when Timon talks of poison hanging "in the sick air" (Timon of Athens, IV.iii.111).
But the sense that causes most confusion with modern usage is that of "needing cure". In All's Well That Ends Well (IV.ii.35), Bertram tries to seduce Diana, saying "give thyself unto my sick desires".
Today, this would mean "revoltingly unpleasant". But Bertram means only that Diana can cure him. And this is the required meaning when Portia says to Brutus, "You have some sick offence within your mind" (Julius Caesar, II.i.268).