mildew

I have a very old adirondack chair, made of cedar. It was sort of old when I got it 20 yrs ago. I decided that maybe I could paint it blue to add a bright spot in the back of the garden. My plan was to clean, prime, paint, polyurathene. I started the first step today. I bought Olympic deck cleaner w/out reading it. Says it must be used on flat surfaces, leave on 5-10 min, no scrubbing, then rinse. I ended up scrubbing anyway. Got it fairly clean but I don't think it got all the mildew because of all the slanting boards. I may try a second cleaning, but stilllllll.....
So, does anyone have anything to suggest? Is this a stupid project? Is there something I'm forgetting? 

Kitty
neIN, Zone 5 
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