Mix the water and salt to make your brine. Make sure salt is dissolved.
Set aside a few washed whole cabbage leaves for later use. Chop remaining cabbage. Slice beets and carrots. Let the vegetables soak in the brine for a few hours or overnight. Cover them with a plate to keep them submerged under the brine. If needed, make more brine to completely cover the vegetables.
Chop garlic and peppers, then mix together, or mix garlic into chili paste if using that. Kimchi is supposed to be a spicy dish, but you can tone down or omit the heat if you need to.
Drain brine off vegetables and reserve it. Taste vegetables for saltiness. If they are too salty, rinse them. If you cannot taste salt, then sprinkle on some salt and toss. Mix brined vegetables with garlic and pepper mixture. Stuff and pack it into clean quart-sized Mason jar. Pack it tightly, pressing down to make sure the brine rises over your vegetables. If needed, add some of the reserved brine. Weigh the vegetables down with the reserved cabbage leaves by folding them up and stuffing them into the jar on top of vegetables. Screw jar lid on, but not completely tight.
Set jar into a baking pan to catch any juices that may escape during fermentation. Set in a cool, dry place out of sunlight. Taste vegetables daily; in about a week they should be ready to eat. Remove the top large cabbage leaves. Store in refrigerator up to 2 weeks.
NOTE: While optimal fermentation temperature is 76° it is not necessary. Depending on the temperature conditions in your household it could take longer, if cooler, or less time to ferment if warmer.