Leftwood said: Consider how soap cleans oil from your cooking utensils...
And if its cleaning grease from your dishes, what is it doing to the waxy coating on plant leaves?
I try not to use Neem. Its too temperamental. White flies are tough. If its a little infestation, I use lots and lots of sticky traps. For big ones, you might have to resort to a systemic but then you have to decide which is more important: bees and butterflies or the problem plant. Of course, if you use a systemic on something you want to eat, .... your worries will be over.
I would be tempted to use the Neem because it is a systemic and because it won't kill you. Read all the warnings and find a new sprayer. Neem takes a few weeks to work so your best plan is to pick the perfect day/time, spray and be patient. Don't repeat.
I know L.A. never really has a cold season but my plan with using Neem is to use it early, before whiteflies appear - preemptive. It lasts a few months.
Good luck!