This question, like many older stories, postulates a sudden appearance of an enormous hazard. This is unrealistic and dated. I've watched semanars from SETI etc. and have learned what kind of surveying has already been done, how much better we can spot something like that now, and the amazing instruments kust around the corner.
Jules Verne was "prophetic" (or more appropriately, prophétique) because he kept up with scientific and technological progress.
Any near-future killer asteroid story would need to consider technology like the LSST which will automatically scan the visible half-sky every week, with particular notes on anything that changes.
I asked on Astronomy SE, and it's pointed out that a mere 3km comet (C/1996 B2 Hyakutake) would have been spotted 8 years out.
All things being equal, a 60km "supernucleus" would appear as a disk 400× the area. I don't know if the brightness figures in the analysis include any coma at that distance. But, the surface outgassing and general mess it leaves behind will also be 400× larger, while the irradiation falls off with distance... if it's twice as far out the irradiation is cut by 1/4, so the coma would only be 100× and further 1/4 the brightness from our viewpoint, or still 25× brighter than the object reported on, C/1996 B2 Hyakutake.
The comet moves fastest at perihelion, and much slower elsewhere. So doubling the distance will more than double the inbound time.
In short, any such object will be seen decades before they arrive.
The documents on C/1996 B2 Hyakutake state that "it's easy to push around". In contrast to the previous answers here, astronomers said that a 3km body was easy to push? The outgassing acts as a rocket with significant ΔV on the comet. The orbit of B2 Hyakutake was changed significantly, such that its previous visit was 17000 years ago and its next will be in 70000. The comet slowed down significantly, all by itself!
This suggests that directed energy would be very effective, adding to that effect. It will also be something of a wild ride, with nobody being sure just where it will wind up, with or without intervention. Slowing it to the point where we are sure of a miss, or heating the top or bottom to push it out of the plane, would start long before it's visible without a telescope.
I suggest the following elements for a more prophétique story set in the next 50 years.
When these instruments come on-line, a very large comet is seen 20 to 30 years out, with concerns of a possible impact.
The orbit is uncertain because it is self-powered and chaotic. Experts don't agree on models. After 2 or 3 years, watching all the (smaller) distant comets provides more certainty and experts agree that the risk is real.
Politicians deny any risk, and eventually a large part of the uneducated population does, too. After all, it's 25 years from now; not our problem.
Somehow or another, work does start. Maybe "other" nations step up and develop huge mirrors. The effort and the mirror technology itself turns out to be a huge boon to the economy and technological development, just as Apolo was for the U.S.
World power and economies shift. Work in other changes such as the effects of climate change 30 years from now, rise in computer power, superconductors, and limited nanotechnology.
Development of the superconductor-based large structures might be a major tech boon. Practical superconductors that work in ordinary and even hostile environments, and flux pinning as a construction technique can be applied to industry on Earth, too.
In short, the armageddon turns out to be a genesis instead: that which doesn't kill us makes us stronger. When the comet passes, large chunks are broken off to keep, and humans set up a manned base on what's left to ride it off to interstellar space, using the mirrors to boost it up to hyperbolic speed.