(UPDATE) - while checking out the numbers in the 'Why?' section below, I've begun to think that, just maybe, the damage might not be so extreme as to force us to do, well, anything except prepare for a sizeable tsunami.
Ten days is not enough to organize and launch any kind of space mission. It can be tried, but in all likelihood it will fail abysmally.
It is not even enough to prepare and launch a missile defense.
Aiming lasers into space cannot be done from Earth, and it wouldn't avail much anyway with the available power.
The only last-ditch, partial and insufficient defense that I can envision is preparing several high-altitude, hopefully unmanned (probably manned by volunteers) jets, and deploy and launch them so that they are between the Earth and the incoming RKKV (actually HVKKV) projectile. Ignoring most safeties and flying a suicide mission with no regard for fuel or engine endurance, a MiG-25 or MiG-29 can reach from 30 to 50 kilometers ceiling.
As an alternative, stratospheric balloons can be sent even higher (and it could be done in addition to the jets), even if they would be much more difficult to aim reliably. However, quantity has a quality all of its own.
Once the projectile is in active range of the balloons and/or jets (for the double mission, there are EMP considerations), onboard fusion bombs are simultaneously detonated through a radio command.
Even more EMP-hardened (okay, okay. Make it "hastily wrapped in aluminum foil and copper mesh") fusion devices will be deployed on the surface, aboard large barges.
The damage to the ecosphere will be negligible anyway against a Chicxulub event, and there is the possibility that the projectile will be fragmented or disrupted enough that the last 30-50 kilometers of air (and the last kilometer or so of water) will be the death of it.
After all, the projectile is more or less just as big as the USS Gerald R. Ford, even if it's ten times denser due to it being solid metal. Just as well, I don't see it faring so well against a multiple multimegaton warhead impact. Granted, the timing will be hairy since the projectile covers fifteen kilometers in a single millisecond, but we do have very precise timers, and very fast proximity detectors.
The projectile will still impact with the same energy, but if the force is distributed enough, perhaps a catastrophic tsunami is "all" that will happen. We'll lose a good part of South Africa, Madagascar and Western Australia, and I don't want to think about the climate or the effect on the Antarctica ice sheets, and the seismic aftershocks will be a nightmare - I expect several faults to go active all at once. But it might still be better than nothing.
The weakness of the Kinetic Kill Weapon is that its penetration depth isn't really related to its speed, but mainly to its density and size. At that speed, rock behaves like a liquid and the resistance it offers is purely its inertia.
That's why "rods from the Gods" are, actually, rods instead of spheres.
If the thing is a kilometer-length rod of depleted uranium, density 19 against about 3 for rock, it will penetrate twenty kilometers of water (or six kilometers of rock) and create a conical explosion that will, in turn, open its way a little deeper and devastate a circular area of the upper crust.
If we turn the thing in two half-kilometer fragments, tumbled 30° from the vertical, the impact will penetrate only at most two kilometers of rock. There will be two smaller explosions, but the destructive effect should be more than halved.
If we succeed in fragmenting and weakening the impactor to the point that the following tenth of a seconds sees the fragment further explode against the atmosphere, there will be a colossal fireball and more widespread, but less intense destruction.
While absolutely devastating, the KKV is still going against a planet. So it is unlikely to wreak even a tenth of the havoc of the Chicxulub impactor, because as said above, it hasn't the necessary size, shape, or volume.
This begs the question, why did whoever sent the KKV send it at all? Why not send a probe to deflect a suitable asteroid, or at least a fleet of smaller impactors, to spread the very localized overkill into a wider "sufficient kill"?
It is still possible that some time before impact, and still out of range of any possible Earth defenses, the KKV will separate into a couple thousand "rods from the Gods" designed to knock a whole emisphere back to the Stone Age; or half of them might slow somewhat, and give their regards to the other emisphere as well.
If that is what's going to happen, then that's what it's going to happen - there is no time or resources to do much of anything, and any evacuation of coastal areas will be useless (Kerguelen isn't going to be the sole major impact point).
But another possibility is that the KKV is just a delivery vector, and that it is designed to do something more than blast a small section of the crust all the way to high orbit. Whatever it might be, it would just have to be something that's not to be accomplished through simple kinetic energy; there must be another component inside the object.
After all, either some guys expended a frightful amount of energy and resources to deliver a blow that in all likelihood won't do very much at all, or they believe it will be enough, which means they know something we don't.
Again, whatever more it is the thing is meant to do, it's not at all certain that it won't do it from very much out of range.
Still, delivering some megatons of nuclear blast at close range might make the difference between a disastrous but limited impact and something far more ominous.