Afik, there's a very abrupt limit on the size of an asteroid where if it's larger it will "touch" the ground and if it's smaller, it airbursts.
So if we could split the asteroid in 10, maybe we go from "completely destroy everything within a 100 miles radius" to 10 Chelyabinsk events.
'Impeding danger' means weeks or days at a minimum, so you have to use a nuke. It would be near moon's orbit.
If it's a bolide, it's in the atmosphere, moving faster than any missile. Its like trying to stop a bullet after it intered you epidemis - you already lost.
[0] I'm aware one of our best bets here would be a proper survey and sensitive enough telescopes/satellites actively looking for these kind of objects. And that there are some projects and programs along these lines already. It'd seem there's some more work to be done in that area though.
The asteroid needs to be pushed with a nuke, or another method, while it's still in deep space, months or years before it would strike. It would be many times beyong the orbit of the moon, millions of kilometers.
ICBMs are for ground targets, their "reach" is a thousand kilometers or so, they have no navigation or tsrgeting for space.
Instead a spacecraft like Rosetta would to be equipped with a nuke and launched on a normal space rocket like Falcon 9 or Ariane or Soyuz. We only need to produce a couple of them in case of launch failure, we don't need thousands of them like we have ICBMs. The launch would not look anything like an ICBM launch, it does not come back to earth.