Kaseki Sousei Reborn II: Monster Digger (Starfish, 1999)
An odd little mashup of a roguelike and Pokemon. You create the creature you fight with by merging three different fossils, although only the head has any visible effect on the silhouette of what you're generating. Seems like there'd be a lot of room for customization over time.
Katou Ichi-Ni-San Kudan: Shougi Kyoushitsu (Culture Brain, 2000)
A set of about 150 shogi puzzles. There's not much more to say.
Kawaii Pet Shop Monogatari (Taito, 1999)
You seem to be running more of an animal hospital than a pet shop. People come by throughout the day and leave their pets with you, and then you can interact with them after closing. I didn't have time to try to translate much of anything, so I can't comment on the story, but the few music tracks in the game are quite good.
Kawaii Pet Shop Monogatari 2 (Taito, 2000)
A classic "more and bigger" sequel. The most notable change I noticed in my brief time with it was that there's more music now, but at least in the case of the town theme they aimed too high and it's not good at all. It has a few different rhythms going on at once that don't work together.
Keep the Balance (Karma Studios/JoWooD Productions, 2001)
You need to bounce the little creatures below up on to the scales so that the weights match. If two matching creatures are in adjacent spots, they disappear. It feels more like a minigame in a third grade edugtainment game than something that should have released standalone.
Keibajou he Gyoukou! Wide (Hect, 2000)
I believe this is software meant to track your horse racing bets rather than anything even pretending to be a game. It also has a glossary of horse racing terms.
Keitai Denju Telefang: Power Version and Speed Version (Natsume/Smilesoft, 2000)
A Pokemon knockoff where all the creatures are attached to phone numbers. The only difference between the two versions seems to be which monster you start with. From there, it's pretty shamelessly Pokemon, but with some interesting changes. You can choose which mon to send out at the start of battle, and defeating a wild mon adds it to your party. Toss in some good music and you've got another game for the list, at least for curiosity's sake.
Kelly Club (Vicarious Visions/Vivendi, 2001)