That's the problem and I think the only place coolant can leak to gearbox is at the waterpump axle. There should be somekind of O-ring thats propably worn? Is is possible to fix it without having to remove the whole sidecover of the engine? Is it enough to remove waterpump cover and waterpump and change the sealing(O-ring) through that? Also if someone knows if coolant an leak to gearbox any other way please tell me.
Remove the side cover and replace the oil seal and water seal on your KDX, pay attention to the way you remove the seals, if the KDX has some hours on it, replace the bearing also while you're at it. Check the impellar for nicks and corrosion on the shaft. You will also have to replace the gaskets.