Big Mistake: Power Steering Fluid top up with Brake Fluid

#1
So as a newbie with car repairs was expecting to make a mistake, but this one is for the books. After making the mistake of topping up with some brake fluid and trying to flush everything the next day, I realized the power steering pump gaskets are gone (loud streaking). So am in the process of trying to rebuild the pump with new gaskets. What I don't understand is why after 3-4 flushes with at least 2 1/2 bottles of DextronIII the fluid is still bubbly/fizzy. Can someone please advise how to do a complete flush of the power steering fluid? Your help will be greatly appreciated. Car is a 2003 Corolla S with 170klms.
 
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