Well, you'll have to stop somewhere; no one flies nonstop from LAX to S.Africa. If JFK is what you are trying to avoid, you can take Delta through Atlanta, or you can take a European carrier like BA, Virgin, Air France, or Lufthansa non-stop from LAX to Europe and change planes in London, Paris or Frankfurt.
If it were me, I'd spend a night in London en route, especially if you are flying in Coach. It takes two days to get there, anyway; if it were me I'd prefer to take a rest in the middle of the trip and get a decent sleep in a real bed.
BTW, I work with a lot of people who travel to Africa frequently; most of them prefer the London route.