I have installed this kit on a number of toilets. It should work fine on your Kohler.
The gasket under the toilet tank that has the three holes is there to in part support the tank. It's not so much about sealing as supporting. If you haven't tossed it, you might think about integrating it into your setup if you can to make the tank more stable, if you can't get it there yourself.
And preventing leaks with this kit isn't so much about tightening the tank to the bowl as it is other things.
First, make sure as Jim said that the hardware in the tank has a bolt head over the rubber washer, and that the metal washer is outside the tank. On this kit, they just give you one washer and one rubber washer, so you put the metal washer and the nut under the hole in the base. If you can actually see water dripping out of the bolt holes, then you may wish to replace the Korky hardware with something like this, that has two of everything per bolt:
What you do is put the rubber washer under the bolt head inside the tank, and make a solid seal by putting the metal washer and nut on the other side of the tank. That way, whatever tension is applied to the assembly from whatever direction when you mount the thing to the base, there is already a solid seal inside the tank. Then you use the second rubber washer, second metal washer, and second nut to secure the tank to the base. This is how pros do it and some manufacturers set it up.
You might also sand a little with very fine paper (before you insert the washer and bolts) around the bolt holes in case there is a little rough area that is interfering with the seal.
If you're not sure that that's where the leak is, check the way you have installed the Korky flush valve. The gasket on the outside (the big rubber thing) does NOT seal the flush valve to the tank. That is the function of the red rubber thing on the bottom of the flush valve on the INSIDE of the tank. There's a chipboard washer that goes between the tank and the big plastic nut on the outside of the tank to make it easier to turn. You should hand-tighten that big plastic nut and then apply about a half-turn with channel locks. If you do that, you won't have a leak at that point (again think about sanding any rough edges around the hole in the tank). The big doughnut washer there just compresses to provide stability and prevent leaks from water splashing back up and out of the hole in the base into which the fill valve empties. It doesn't require much compression to make a seal against that, but if you have a leak between the flush valve and the inside of the tank, it can come out over the top of that doughnut.
If you have installed the Korky kit properly, the tank should be very stable. You can tighten as long as no porcelain is contacting porcelain. You can check this by sliding a piece of paper around under the tank and by examining the tank and base to see where the points of contact are.
The original Kohler parts are not necessary and shouldn't add anything.
You can also call Korky for help at 1-800-LAVELLE. They are nice folks in Wisconsin and happy to help you.