sounds like the water pump seal, make sure to put it in right,and flush the tranny out with kerosene or something
the kdx are great bikes just maintain them like all bikes and it will last
the kick starter sprag spring is weak replace it its not hard , get a manual you will not regret it
the rings are prolly wore,check the piston to cyl clearance,and ring gap
hone it lightly to remove the burnt crap in the cyl
check the reeds if there weak it will run rich,and not want to start, keep the vent tubes clean ,and clear,also check the crank case vent tube, you might want to add a 1 way check valve so if you ride in deep water it will blow out pressure but not suck in water
check the float level by the book
do a fresh top end compression check ,and take notes for the break in ware
seal all the gaskets,and wires,and breather, use a good air filter
i like klotz oil it is really good oil ,and burns clean