For fermentation in general, let me give you a few tips that may be tripping you up:
Don't wash ANYTHING in chlorinated water before going in the jar.
The salt type and granule size is very important.
Always calculate salt by total weight of the liquid + contents going in the jar
NEVER add chlorinated water to the vessel. NEVER EVER EVER
Some tips for Kimchi specifically:
Smash the garlic and let it sit for a few minutes before chopping or adding
Get the finest granule of pepper "flakes". Not the large flake kind meant for soups or stews. You want powder, practically. Deep red, and about the grain of table salt at max.
Mix your salt, sugar, pepper flakes, garlic and onion FIRST before adding them to your cabbage. A little bit of soy sauce or spring water for moisture. Let the mixture sit a good 30m to bloom before rubbing into the cabbage.
RUB THAT MIX ALLLLLL OVER EVERRRRRYTHING. When I say everything, that means every little nook and cranny of the cabbage. Squeeze the cabbage a bit while doing so to help release some moisture into the marinade.
Get a muddler to help smash and bruise the cabbage as it's going into your jar. You really want it packed tight, and smashing it as you go releases even more moisture into the mix, so by the time you've filled the jar, you should have enough volume to mostly cover the top
Get some weights for the top to make sure the last bits added are submerged.
With kimchi, I used very little water (only the amount the recipe suggests). The veggies I use( Chinese cabbage) contains a lot of water naturally and my batch always overflows once fermentation happens.
You should search around, but depending on what your tap water contains you may be able to just use counter top filtered water if you let it stand for awhile. Bottle spring water is a safe bet though, and should have extra minerals in there to give a bit of a kick to the taste. Just like bread fermentation, it's all about the water source.
Yeap, most people are gonna be fine with regular chlorinated tap water as long as they let us sit for a while. People having problems with their starters because of tap water is due to their municipality using chloramine instead of chlorine. It's not as volatile as chlorine, so it can stick around in unaerated water for days.