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Assuming nothing breaks (unlikely), you could potentially calculate it by calculating the energy stored in the engine/axle spinning. We know that energy must be conserved, so the energy after hitting the ground must be converted into heat and kinetic energy (of the car), with some energy left in the axle to keep it spinning.
I don't know enough about car parts but it seems feasible to at least estimate an answer given enough information
Good approach. Start with the momentum of the rotational masses, assume no suspension (to make the math simpler).
My guess, a few feet. There's surprisingly little momentum in all those components compared to the mass of the vehicle.
"surprisingly" is a strong word; cars are designed to have as small of a rolling mass as possible, because it's much harder to stop than simple dead weight.