Success I found something useful:
Put the handle back over the handle and tighten the knurled cap, while holding the handle.
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I hate 1 sentence answers, I write too many words, such is life. I also enjoy teaching when I can. Below is my 'real' answer, it is long, I do not intend to talk down to you, simply to provide as much information as I can. I hope it is useful.
At the bottom of the page, is a cut and paste from the manufacture's website.
The first 3 steps of it will allow us to identify your problem.
1 Turning off the water and reliving the pressure is just a good idea.
It helps keep oops from being a real problem.
2 Unscrew the cap. This is what happened to cause your problem.
3 Handle and Cap come off together - You know this, the question is why.
Hopefully the knurled cap will turn without much friction while holding the handle.
If you need a wet paper towel, or a rubber jar opener as they mention, we have a problem.
There will likely be 'gak' (to use the technical term) inside the handle and cap. A collection of soap and other interesting things.
Lots of hot water, q-tips both the cotton and the stick part, toothpicks, rubbing alcohol, vodka (inside the handle or inside of you) and anything else that will not scratch the metal are fair game.
The part sticking out from the faucet, where the cap attached is the 'cartridge'. It is the working part of the valve it controls the on/of and mixing of hot and cold water. If your faucet were leaking or dripping, this is the part that would need to be replaced.
The top of the cartridge has a tapped hole, this is where the nut screws in to hold the handle onto the faucet. There will be gak on top of the cartridge, and you want to get rid of it too. If the cartridge top moves or turns, that is fine, but it should be very tight and only move with real effort.
There is, or at least should be a flat spot on the cartridge. This should line up with an opening in the cap end of the handle. This serves as a grip area for the handle and the cartridge. It also makes sure that the handle will not spin endlessly and have no effect on the water flow.
The flat spot on the cartridge should be flat and not show signs of wear. If it is brass this is unlikely, if it is plastic, it could show signs of unnatural rounding. It the faucet handle had a 'sloppy' feel to it, this is the problem area. The only solution is to replace the cartridge.
At this point, slide the handle back on, get the flat part of the cartridge lined up with the mating part of the handle, and screw the nut on until it is tight. Tight in this case is very tight, without tools. A wet paper towel, or rubber jar opener can be use to get a grip on the cap, but not more than that.
Move the handle on and off a few times, the cap should turn with the handle, and not just sit there.
Turn the handle to what you think is off, crack the valves a bit, to make sure that things still seem leak free. Then open the valves, test the handle again, to make sure there are no leaks and the cap is turning with the handle.
If all goes well, celebrate with a glass of water. :)
A couple of notes, since you are digging around there anyway, and the faucet is a few years old, replacing the cartridge is not a bad thought. Directions from the manufacture's web site are below.
Cartridge come in a hundred styles, it seems. You can find 3rd party ones at big box places, and this might be a bit high end for your local hardware store, but I give them all the business I can. Or you can order the oem one from the mfg, or somewhere online. With faucet cartridges, you get what you pay for.
Check the cap a few times over the next couple of weeks, to see if it is unscrewing itself again.
If it is, and you feel comfortable with it, you can remove the cap and use 1 or 2 drops of - Purple - Loctite. If you still have problems, or insist on more power the - Blue - would be ok.
Best of luck, I will do my best to answer questions, or learn from error that you find in my answer.
Below is a cut and paste from: http://products.whitehauscollection.com/home/technical/
Cartridge Replacement
- Shut off water under sink. Open handle to relieve pressure
- Unscrew cap by hand, rubber jar opener or strap wrench may help
- Handle and cap come off together
- Unscrew lock nut on cartridge
- Lift cartridge out of body and inspect inner body for Debris before installing new cartridge. (A tiny bit of Silicone Grease on bottom of washer is ok)
- Small nub on bottom of cartridge will fit into little climate body – replace lock nut, make snug.
Best Answer
That is the stem, but looking at your pictures it appears the faucets are 1/4 turns. So if you can turn it more that 1/4 of a turn and not get water, you need to turn it the other way. If the other way doesn't work you'll need to replace it, because that's all the knob/handle would do. You may have loosened it and need to turn it back to where it was to start. Guessing from photo on the 1/4 turn, if they aren't you may need to just turn more or try the other way.
You can also remove the screw holding down the other handle and use the same handle on both.