The story never actually says, one theory is since Jack found out he was brainwashed, the mind control lost its effect. It is also possible Andrew Ryan simply had no desire to harm his son.
It could be he left Jack alive, so he could send him to kill Frank Fontaine, motivated by being sent to kill his own father & lead along by him.
Speculation / theory: (Please note this is the theory of a contributor to the Wiki & should not be taken as fact.)
Andrew Ryan’s death might have been faked. It was maybe an animatronics dummy from Ryan Amusements, with some programming that made Jack believe it was the man himself & Frank Fontaine was stupid enough to fall for it too.