The question if you can get a favorable relative speed is a question of fuel and burn times.
Getting there to hit it, is easy (besides the actual hit) but you likely have to break again to match the speed again, to come down to the 9km/sec you mention.
Not challenging your numerical figures, however a nuke hitting will not even transfer 1 pro mille of its energy to the asteroid. So if your calculation was correct, the speed change of the asteroid is in the range of mm per second, not metes, likely even less. Hold a hand besides a flame of a a candle or a lit bulb. Only the radiation on the side where your hand is will affect your hand or in the asteroid example the asteroid. The rest of the explosion in the "sphere of the light bulb" just goes into space.
OTOH if you indeed manage a direct hit, preferable with an explosion 'under ground' then only that part of the explosion that ejects matter will alter the course of the asteroid.
The rest of the energy will simply heat up a crater of molten /evaporating 'stone' which will only change the momentum minimal ... especially considering the rotation of the target, which will spread the acceleration of that radiation equally.
In other words: the transferred energy will be 99% heat, and only one percent momentum. And depending on the 'hit scenario' the transferred energy will be probably less than 1 pro mille, so bottom line only 1/10000 of the nukes energy changes the impulse/course of the target.