When you are going for kotegaeshi, and someones hand is stiff as a board, and the guy is stronger than you...how do you 'break' the stiffness to get the bend needed for kote gaeshi?
Much has already been said. When people resist, there are two things that I regard as particularly important:
1 Extend uke's arm forward, before making the kotegaeshi twist. Like mentioned earlier: tori's movement is from uke's elbow toward the wrist, and pushing on in that direction.
2 Keep uke's hand right in front of your center, and use your whole body in the twist. When uke is very strong, you need to keep the hand very close to your center, maybe even hold it against your belly.
Also pay attention to how you grip uke's wrist. Make that grip a squeeze to sligthly separate uke's hand from the underarm. That will weaken uke's resistance power significantly.
A completely different solution is to get things going, making uke stumble forward, and then do the kotegaeshi more like a kokyunage than a wrist twist.