No one here had the answers, so I am going to produce them myself.
The ballast I bought, a Hylux 2A88, cost me around $48, in contrast to the multi-hundred-dollar Stanley ballast Infiniti dealers sell and the $270 aftermarket ballast an online parts company sells. It works fine. It's potted like crazy; I would be stunned if water ever got into it. The Stanley is held together with screws, and the interior of the box is full of a soft, mushy substance that resembles polyurethane. You can remove it with your fingernails.
You have to get the old light out. Nissan designed it very poorly, so it's hard to work on. The owner's manual drawings were apparently created by a chimp, so they're not helpful. I did the driver's side.
You don't actually have to remove the car's grill, the way the directions say. You just have to remove the coolant reservoir and the air cleaner, along with a few little items that get in the way. You should cover the paint around the light with a double or triple layer of tape to prevent scratches. After that, you disconnect the two modular power connectors that provide DC to the light, and you can wiggle the light out of the car.
The ballast is under the light assembly. It's held on by three screws, which you should keep and reuse. Take them out, disconnect the ballast, remove the ballast, cut all the wires leaving it as close to the ballast as you can (to conserve wire), and throw the ballast out.
If you can work on the wires without removing the bulb and socket from the assembly, you should do so, because reinstalling the socket is very hard. If you have to remove the socket, take pictures first so you can get it back in.
Don't listen to anyone who tells you to use a new light socket that goes with a new ballast. New sockets do not fit the Infiniti light assembly. You will never get it installed. Use the old one. It has nice long wires, and you will need long wires when you install the new ballast.
I removed the shielding that covers the wires going to the socket, based on the fact that newer headlights don't use it. I hope I don't regret that later.
I did have a new socket on hand, and it was useful for making an adapter. I cut the socket off, leaving a female connector that matches the Hylux, plus two leads. I connected this to the leads from the old socket, allowing me to use the old socket and keep the long wires that came with it.
Here is the new socket on Ebay:
http://www.ebay.com/itm/262013504979?_t ... EBIDX%3AIT
The old socket assembly--you have already cut the wires where they meet the ballast--has three leads. The Hylux ballast is made for two leads. You have to join two ballast leads to three Nissan socket leads. This means you have to join the two negative socket leads and solder them to the single negative lead on the new connector.
The red Nissan wire with the fluffy insulation goes to the center of the headlight socket. The other two old wires go to the rim contacts. You need to solder them together, so you have one red wire (+) and two joined wires (-) that you can solder to the wires that come from the new connector. The new wires on the setup I was using had a red positive wire and a blue negative wire. Polarity is important. That's what I read, anyway.
The connector on the Stanley is obscure. The Hylux uses a much more common connector. To make it work, you have cut the female connector off of the power supply going to the headlight (the Nissan connector) and solder in a new adaptor, taking care to cover everything up with layers of heat-shrink tubing. If you can't work solder and heat-shrink tubing, you should really learn, because it means you're hopeless.
Here is the connector I used:
http://www.ebay.com/itm/121402552709?_t ... EBIDX%3AIT
It says "Universal 9006/9005 HB4/HB3 Fog Light DRL Female Pre-wired Connector Socket."
When all this is over, you will have the old Nissan bulb and socket, the old Nissan wires coming away from the socket, a joint where you joined three Nissan wires to two wires salvaged from a new socket, and the female connector (from the new socket you cut up) that goes to the Hylux.
The new ballast has a long wire and ring for a ground, but this is not needed for the Infiniti, so cover the ring with insulation and fasten it out of the way with a cable tie. I suppose you could attach it to something, but I was told it wasn't needed.
Make sure you insulate everything well on this side of the ballast, because the output voltage is very high.
If you had to remove the headlight socket, put it back in. Run the adaptor and new wires back from the headlight socket and out through the hole where the Stanley used to be. Now you're ready to install the new ballast.
The ballast I bought has three screw eyes for mounting. They don't line up with the eyes in the Stanley, so they don't line up with the corresponding holes in the headlight housing. You will need to make a couple of adaptors from scrap metal. I used 1/8" aluminum. You drill holes to attach the aluminum to the ballast, and then you drill holes for the screws that go through the aluminum into the housing plastic. I tapped the holes for the ballast (in the new aluminum) and ran 3/8" #8 screws into them with Loctite. It saved space compared to nuts.
The new wires you made for the socket will have to exit through the hole that used to mate up with the Stanley. There will be some space beside them where water could conceivably get in. I suppose you could pump it full of silicone sealant. I didn't bother.
The new wiring will be loose and floppy. You will probably want to restrain it with cable ties.
When you're all done, you will have a Hylux ballast screwed to the old headlight assembly, along with one connector that has to be attached to the same 12V power wires that used to feed the Stanley. You created a new connector (female) on the Nissan side, so carefully push the headlight back into the car and attach your new 12V female connector into the Hylux male connector.
Replace the modular connectors, reinstall everything, and you're good to go. The light comes on instantly and looks just like the other one.
After all this aggravation, I have to say that I really regret recommending this lemon to my dad. The dash screen died. The sunroof died in the open position. The car eats batteries because it has electrical leaks no one can find. Replacement headlight ballasts cost something like six hundred bucks each, new, and they suck. The clear coat peeled. The electric windows failed. The upholstery and interior cracked and wore prematurely. It's too bad Infiniti didn't put this nice chassis and engine under a real car instead of a sickly piece of junk.
I still have to fix the dash screen, and that means finding the right part, in working condition, and removing the dash to put it in. The brains at Nissan thought it was a good idea to link this weak, unnecessary part to the AC and stereo, so the car is useless without it, unless you enjoy playing blind man's bluff with the temperature and sound. Next time, Lexus.