Crude drawing of how you need to cable the fans up. I'll leave it up to you how you do it. Splice using crimps, or solder with shrink tubing to cover. Give yourself plenty of extra length on the cabling, you'll need more than you might think.

As written the 3-pin to 2-pin adapter in the parts list is going to be cannibalized, you only really need need the single 2-pin connector, and you will discard the 3-pin connectors.

You could leave the new fan's connectors attached, and rig this up so if one of the new fans fails, you only need to replace the fan without cutting wire, but then you will need to two 3-pin to 2-pin adapters per cube.

Once you have the two fans and connector assembled, test them to see if they are connected correctly by testing the resistance between black and red at the connector using a voltmeter. If you have got it cabled right, it should not be an open circuit, there should be some resistance there.

Of course, if you have got a 24V DC supply, you could test them that way too, or even hook them up to the cube, plug it in, and turn it on (it will have to have water in it, and be turned right-side up, and have the control board plugged back in.)