Have two large bowls handy to place cut vegetables. It is helpful to have a third bowl available.
Remove the damaged outer leaves and wash cabbages thoroughly. Cut into quarters then slice down to remove the inner core. Discard.
Wash the kale and remove the stems dividing them down the middle. Stack the two halves of the kale on top of each other. Slice across the stacks in 1/8-inch strips. Divide in half between the two bowls.
In a food processor with a shredding attachment, shred the carrots and divide them between the two bowls. Do the same with the daikon. It is not necessary to clean the machine between uses because they are all going into the same bowls.
Sprinkle 1/2 to 1 teaspoon salt over each bowl of veggies and massage them until they begin to soften and become moist from the water content of the vegetables leaching out due to the osmosis action of the salt.
Lay each cabbage quarter on its side. Doing one at a time, cut across the closest edge to form thin ribbons of cabbage, about 1/8-inch thick. Transfer half to each bowl. Do this until all the cabbages, both green and red, have been sliced and divided between the two bowls.
Sprinkle the remaining salt over each bowl, tossing the various vegetables to combine. It is helpful to use a third bowl to mix some of the veggies from one bowl, salting them at the same time, then adding more veggies until they are all combined and salted. Massage the salt into the veggies. They will quickly become damp from the fluids leaching out of the veggies into the bowl. Return the veggies to their original bowl. You will notice that they no longer have the volume they initially did and are much limper than before. Do the same with the second bowl of vegetables.
Allow to rest for thirty minutes to one hour, or until the vegetables are limp and a noticeable amount of liquid is pooling at the bottom of the bowls.
Transfer each bowl of vegetables to a sterilized half-gallon canning jar, packing the veggies in as tightly as you can. Place a glass fermentation weight on top of the veggies if you have them. With a clean, damp cloth, wipe the rims of the jars so the caps fit without being obstructed by pieces of vegetables.
Close with a canning lid. (You may use smaller jars but it is more convenient to ferment them for a few days in the larger jars, then transfer them to the smaller jars just before the fermentation period is over. They require a lot of monitoring for the first four or five days.)
Place the jars on a plate or in a bowl with the canning lids slightly loosened. If you have a seedling mat, it is very useful to place the plates with the jars on it. This assures the temperature is high enough for proper fermentation to take place.
Leave the lids slightly loose so that air can escape while the fermentation process is going on. It is normal that liquid from the vegetables will build up to the point that it spills over the sides of the jars into the plate or bowl the jars are sitting on.