How long would it take for Mars to cool after asteroid impact?
It would be interesting to read peer reviewed scientific papers that discuss the feasibility of "restarting Mars' core" via an asteroid impact. Some links would be appreciated.

I am very sceptical that it could be achieved in such a manner. Small asteroid impacts would change the topography of the surface by creating new craters and spraying crater ejecta over the surface.

Impacts from extremely large asteroids would only create larger craters or possibly blast out large chunks of Mars and even destroy the planet: such as a collision with Ceres.

One of the flaws I see in trying to "restart the core" is how to keep a solid inner core within a molten outer core, while at the same time creating relative movement between the two.

The Earth's magnetic field exist because its solid inner iron-nickel core and the metallic molten outer core move relative to one another creating a geodynamo. Without a solid core and relative movement between the two the only thing that will be achieved will be a molten core without a magnetic field.

If it were possible to remelt the core of Mars, a magnetic field would not be created by simply remelting the core.

The other flaw I see is that if it were possible to remelt the core, the core will expand due to thermal expansion. This will then increase ground stresses within the overlying rock which will result in cracking of the overlying rock and the generation of Mars-quakes.

Another flaw I see is concerns the energy required to remelt the core. A significant volume of core would need to be remelted. That would require a lot of energy over a prolonged period of time. The effect of an asteroid impact would dissipate quickly.