"Ille syk" sounds like something you would say in an Østfold dialect.
Kvinnen ble illest fordi hun hadde syk rhymes.
You could also say "Kvinnen ble (så) veldig syk"
English grammar says that when "so" is used in this way a dependent clause should follow (a rule that is widely flouted). Is Norwegian grammar similar? Example: The woman got so terribly sick that she had to leave.
But it's also very common to use it as here, such as, I'm so happy, this is so crazy, that's so interesting.