It doesn't necessarily need to be diverted entirely. Simply breaking it up into smaller components significantly reduces it's impact force.
Smaller pieces burn up more readily in the atmosphere and create smaller impacts (I believe exponentially smaller)
some of these asteroids we only have days warning. that's the scary part. To divert an asteroid you need to do it before it comes near so it has a long path to begin drifting in a new direction. ie it's only gonna nudge it slightly. Slightly adds up over many millions of miles, not so much when its a week out.
Now you're begging the question of how hard it is to break it up!
Significantly harder than diverting it slightly.
If you know find out years in advance then you barely have to change it's velocity to make it miss Earth. Probably a kinetic impactor would be sufficient, otherwise thermonuclear warheads exist.
Capable with current technology if intercepted early enough. I think most people visualize an asteroid hitting earth like a ball hitting the ground when dropped at arm's length. In reality, orbits of celestial objects are elliptical, and they're constantly moving. A small force, like that produced by an ion thruster, applied to an asteroid 15km in diameter for months or years would be enough to change it's orbit such that it wouldn't impact earth.
when did we learn it was heading our way, and how far away is it at detection? hours, days, months, years before impact? that's key things to know.