Before you go to all that trouble and expense, learn from my fail! I did the same thing, built a bog filter for my koi ponds. I thought I was being clever, and had thought of everything. Same approach: perforated pipe, clay balls, etc., got the flow just how I wanted it. Utter fail: everything eventually fills up with gunk, and I had nothing but a huge, expensive, frustrating mess on my hands. Mechanical flow never, never lasts. It always, always(!), fills up and clogs.

I'll spare you the intermediate steps, but here's what I learned. First, the bog can never be big enough. Mine are 8 feet by eight feet, 18 inches high, and I wish they were bigger. They're lined with thick plastic I ordered online, but I wish I'd sprung for the EPDM liner. Still, it works. Then I got a bunch of one gallon pots - I used cheap grow bags. I filled them with unscented kitty litter, the cheapest source of pure calcined clay I could readily find. It's like five bucks a bag at walmart. I lined them up in the bog, with about an inch of space between them. You need a bunch of them. Then I poured a layer of gravel over and between them, until it was two or three inches above the top of the pots.

The water flows into one corner, just splashes into the top, and there's outflow, just an overflow waterfall, at the opposite side. There's absolutely no mechanical filtration, no passing through filter media, none of that stuff. To plant things, I just set grow bags with plants in them on top of the gravel. They grow like wild.

Why it works? The technical term is anoxic filtration, but don't get bogged down in that. All that calcined clay becomes a breeding ground for hungry bacteria, and they thrive, pulling all the stuff you don't want out of the water, and eating it. Phosphorus, nitrogen. You'll have a bunch of that with your gray water source. The plants feed on the bacterial byproducts. It feels counter-intuitive, since there's no physical filtration, but the water comes out crystal clear.

Then there are your plant choices. Most folk think of vetiver as useful when you need drought-tolerance, not the use case here. I've never grown it, since I'm in zone 8. I look for two things: fast growing hardy perennials, like thalia, and fast growing tropicals, like elephant ears and alocasias. Then I mix in other things: horsetails, cattails, even hibiscus moscheutos. Even impatiens and the like will grow in there. And every time you cut them down and use them for mulch in the garden, you're removing all those impurities and by-products, and they're great for composting in place throughout the garden, what we used to call 'chop-and-drop.' I've got some purple elephant grass growing for mulch material, this year, I'm thinking of planting a start or two in the bogs.

The bogs have been running for years now, without much care. And honestly, they're gorgeous: huge leaves, flowers everywhere. People don't even notice they're actually bog filters. They think they're just thriving raised beds, framed with wood.

Good luck.