We send spacecraft on comparable missions all the time. And it doesn't really take a spectacularly large payload to destroy (yes, destroy) an asteroid a few hundred meters in diameter. 1/2-kilometer-wide Itokawa could be blown into tiny bits which would not recoalesce, via a 0,5-1,0 megatonne nuclear warhead, a typical size in modern nuclear arsenals (in addition, the little pieces would be pushed out of their current orbit).
I know it's a common misconception that "nuking" an asteroid would simply create a few large fragments that would hit Earth with even more devastation, but that's not backed by simulation data. And anyway, even if it didn't blow the asteroid to tiny bits (which simulations say it would) and even if it didn't push the remaining pieces off trajectory (which they say it does), anything that spreads an Earth impact out over a larger period of time is a good thing - it means the higher percentage of the energy that's absorbed high in the atmosphere rather than reaching the surface (less ejecta, lower ocean waves, a broader (weaker) distribution of the heat pulse, etc), the weaker the shockwaves, the weaker the total heat at any given point in time, and the more time for Earth to radiate away any imparted energy or precipitate out any ejecta cloud. If the choice is between 15 Chelyabink-sized impactor (most of which will strike places where they won't even be witnessed) or one Meteor Crater-sized impactor (same total mass), pick the Chelyabinsk ones. 50 10-megatonne meteor crater impactors or one 500-megatonne Upheaval Dome impactor? Pick the former. The asteroid impacts calculator shows the former generating a negligible fireball and 270mph wind burst at 2km distance, while the latter creates the same winds 25km away (156 times the area) and a fireball that even 25km away is 50 times brighter than the sun, hot enough to instantly set most materials on fire.
But that's all irrelevant because, quite simply, simulations show that nuclear weapons do work against asteroids.
What we need is enough detection lead time to be able to launch a nuclear strike a few months before the impact date (to give time for the debris to disperse). There is no need to "land" or "drill" for the warhead. There is no pressure wave; instead, an immense burst of X-rays is absorbed through the outer skin of the asteroid on the side of the explosion, causing it to vaporize (unevenly) from within, especially near the ground zero point, and creating powerful shockwaves throughout its body. In addition to ripping it apart, the vaporized material and higher energy ejecta flies off, predominantly on the side where the explosion was detonated, acting a broad planar thruster.