Insert switches into the plate with the LED window to the back and the pins to the front.

The stabilizers consist of two parts and a rod. They must first be assembled and then added to the plate from the underside.

Thereś plenty of tutorials that get into the details of installing those stabilizers. Here's one:

Kailh Chox stabilizer installation

After making sure all the switch pins are vertical and none are bent out of place, carefully align the plate and switches with the PCB. Slowly and carefully complete that assembly, making sure no pins get bent. Once that is done, the plate and PCB should be a few millimeters apart, evenly spaced across the whole surface.

You may secure the boards together with tape or clothespins until you've soldered enough switches.

Solder each switch onto the PCBs.

Note the console key switches are not within the area covered by the plate and need to be soldered separately after being carefully aligned by hand.

Solder the 24 pin IDC connector to the top of the connector board, the 22 pin header to the bottom and the 2 pin power header to the top.

Solder the second 24-pin IDC connector to the bottom of the keyboard and connect both ends of the flat cable into the IDC connectors.

Assemble the key caps onto the switches starting with the stabilized keys. Don't add the caps to the console keys as that will make it difficult to get the keyboard through the tight console key holes.

Your keyboard is now assembled. Let's mount it into the computer.

Opening the Atari 400 can be tricky if you haven't done it before. There are some great tutorials and videos showing how to do that. You only need to remove the top of the case, no need to fully disassemble the computer.

Once the top of the case is free, carefully lift it and unplug the keyboard ribbon cable from the connector on the motherboard.

Carefully remove the membrane abomination from the top of the case, taking care not to break the two very small pegs that hold it on the bottom. Some bending will be necessary.

Get the new keyboard in place under the top of the case. Getting the console key switches through is usually the trickiest part. Also take care to get the PCB in place on the bottom.

The top will still be floating, but it will be held by the shielding when we reassemble the computer.

The keyboard's backlighting needs to get power from the 400 board since the keyboard connector doesn't have a 5V line. Power and ground taps can be found on the power board.

The black wire is ground and the red one is 5V.

Carefully insert the pin header on the bottom of the connector board into the motherboard's keyboard connector.

Connect the two power lines to the remaining two pins on the connector board.