Nuking asteroids, saving the Earth

>> It makes a much less exciting movie, but the best way
>>would be to attach a solar or nuke thermal rocket to
>>it, fed from asteroidal ices.
>
> Or how about a mass-driver?... one would in this case be
>throwing away the least valuable parts of the asteroid as
>reaction mass, rather than the most valuable.

An alternative which has been suggested is to land tools to bootstrap the
entire operation: metals to make foil mirrors to increase the output of the
facility are easy to make, as are the rails and equipment which are the bulk of
the mass driver track (rolled & stamped metals, drawn wire for cable, and
fused rock-dust "aster-crete/sinter-crete/concrete".
As time goes on, while the thing goes around the Sun a few times (while
crews are occasionally rotated or resupplied with finished goods they can't
produce as needed), the track length of the mass-driver engine lengthens,
driving up the exhaust velocity of the rocket.
There's a narrow race being run between the mission concepts of bring
everything needed to set it up and operate it straight from the cargo hold, or
instead bring more tools to build the dumb bulk parts which are by a large
margin the most of the cargo you'd need to bring to run the facility.