Reddit mentions of StarTech.com 6in 4 Pin LP4 to Left Angle SATA Power Cable Adapter - LP4 to SATA Power Adapter (SATAPOWADPL)

You can remove the existing SATA end and replace with one you crimp on yourself, or buy one of that style like this: https://www.amazon.com/StarTech-com-SATAPOWADPL-6-Inch-4-Pin-Adapter/dp/B00AC8NK8Y

The key to to only get/use ones that are crimped on, the other kind can be safe, but unless you want to use an xray check the tolerances and such it is a gamble.

Awesome, thank you! That's what I needed, more ideas on how to implement this. Either molex or usb or direct pins, one of them is bound to work.

And I do agree, All-in-one PC's are a bitch to repair ... thankfully in the 3+ years of owning this one, this is the first time I've needed to open it up. :)

EDIT: I've just ordered a (slightly complex) solution from Amazon. This all-in-one only has one SATA power connector coming out of the PSU that goes to the HDD, so I'm going to convert that into Molex, then use a Molex adapter that has a three-pin sprout,, connect the new fan to that, and then reconvert the power connector back into SATA, to plug it back into the HDD.

I feel like as long as all of the connections are tight and I keep it all tied down, it should work just fine. It's a bit more complex than it would have been if there was a three-pin connector on the motherboard, but no such luck. What do you think?

Thanks again for your advice.

I ordered this

The "overmolded" type of SATA cable are known to be dangerous, the heat can't escape easily and tends to soften/melt the plastic over time. Eventually the wires inside the connector are bare and come in contact/short.

Ironically, the "crimp-on" or "pass-through" type of cable are much safer, despite the fact that they don't look as nice.

You certainly would not want to pass 60W+ through an overmolded cable like in a mining rig, this is coming real close to the electrical limit of even a good-quality SATA connector and the cheap overmolded stuff can be burned out just from a SSD in some cases.

Molexes are a little better, but the electrical limit is still only about 130W. It's much safer to use connectors with the 6-pin PCIe type connector, these are design-rated at 75W and the electrical limit is ~175W.

I'm adding the drive. None of the connections in the old computer fit the new drive, but I found a molex to ata adapter (now that I know what it's called). Thanks for your help.