As kamukiman noted, I think kimchi would be difficult because the cabbage would already be really soft and may not take in the salt and seasonings very well. I'd also be really concerned that the fermentation won't happen properly and you'll just get slimy rotten cabbage. You could use the kimchi seasonings to make a spicy cabbage stirfry like ipsedixit suggested, or maybe a stew like kimchi jjigae which uses very fermented kimchi (which is usually quite soft, although you would need some sort of acid to replicate the tartness).
If you just need a suggestion on how to use it up and don't have to go asian-style with it, my mom makes a very simple and good cabbage-bread soup which can use up lots of cabbage (and leftover bread and cheese too).
Cheese, Bread and Cabbage soup
Cooking time: 1-½ hours
1 loaf rustic sourdough, baguette or peasant bread (~1 lb), cubed and dried/stale (something with a crust is best. Regular sandwich bread will turn to gritty/slimy goo)
1 head of Savoy or green cabbage (2 to 2 ½ lbs) cored and thinly sliced
1 pound of cheese, shredded (I have used various mixes of fontina, gruyere, cheddar, manchego and many others in this soup. if the cheese would taste good in a grilled cheese sandwich or fondue, it will be great for this recipe)
2 quarts broth or stock (chicken or beef, or any good vegetable broth that isn't overly sweet)
salt and pepper to taste (especially if using a low sodium broth)
Layer bread, cabbage and cheese in 6 to 8 quart pan, alternating ingredients like a trifle, repeat until full, ending with a layer of cheese. Press ingredients firmly to level and pack them into the pot.
Heat the broth until boiling and pour over layered ingredients.
Cover the soup pan with a domed lid or cover with foil dome.
Set soup pan in a rimmed pan to catch any overflow. Bake in a 400 oven until a crusty crown forms, about 1-½* hours. Dip down through ingredients and ladle soup into bowls. Warning it will be VERY HOT.
*Since you have already cooked cabbage you can probably cut the cook time down to 30 min or however long it takes to melt all the cheese and turn everything all molten and bubbly. Because it cooks so long I usually don't even bother pre-heating the oven, but you probably should preheat if you're doing a shorter cooktime.
It's almost like french onion soup, but without the onions. The cabbage gets soft and luscious like caramelized onions, the bread makes it hearty and the cheese makes it delicious. This is also a very forgiving recipe. Have more cabbage, less bread, different cheeses? No problem. I've never made it exactly the same way twice and it's always been good (except for the unfortunate sandwich bread version).
Pickled cabbage, be it German Sauerkraut or Korean Kim Chee, or anything in between almost all start with raw rather than cooked cabbage, but I suspect you already know that.
Chow had a recent recipe for Kim Chee:
Try to dry as much liquid off as possible (salad spinner?) and omit the initial salt-and-soak step, see how it goes, but I'm guessing its going to be pretty mushy. Maybe mix it half and half with additional fresh cabbage to get some texture back into it? I'm thinking Ipsedixit's suggestion might be better, sort of a japanese style.
You might look up recipes for Namasu, subbing the cabbage for cucumbers, but still mushy issues.