First, bend and solder all the diodes. Make sure to solder them so that the black band is on the side of the square soldering pad. I used these pliers to bend them.
As you're soldering the diodes you may find it easier to go through and cut the legs off the ones you've soldered. I'd recommend using these.
Next, insert the headers, long side through the Gherkin PCB, put the KB2040 on it to hold it in place, and solder the headers to the Gherkin PCB. The long side of the headers should be on the same side as the diodes, and the short side as well as the plastic bits should be on the other side. After you've soldered them, cut off the excess on the side you soldered so there's space for the switches.
Now, solder the switches in. Make sure that the side of the plate with the markings on it is facing down. Just about any orientation should work, but some keycaps can be pretty specific, so make sure to test that before soldering all the switches.
After that, solder the KB2040 to the headers. Have it face inwards, and make sure to leave enough space so the switches don't come in contact with it and short anything out (I cut the legs off the two switches above the KB2040 to ensure this wasn't an issue. You'll notice the KB2040 is has two extra pins. Don't connect the pins labeled D+ and D- to anything.
Finally, attach the top plate and bottom plate with the included standoffs. Put standoffs in all the holes except for the two directly next to the USB and make sure that the logo is facing out on the bottom plate. (also disregard me using 3 very different switches in this build, I ran out of other switches). This keyboard has a pretty non-standard layout so I'd suggest looking at the diagrams in the "Setting up KMK" page when installing them.