I can't let this go without pointing out that the question is flawed. (This same flaw in textbook problems crops up on the forum several times a year.)
By asking for 'the force', the question assumes the force is constant over the 1cm. That is clearly not the case. Since nothing breaks, it might be pretty much like a spring, with the force increasing linearly with extent of compression (but with a much lower constant during decompression). The peak force will therefore be about twice that computed by energy/distance.
Most questions like this try to get around the problem by asking for the average force, but that doesn't work either. There are two ways we might try to define average force here:
Average over distance: ∫F.ds / ∫ds = ΔE/Δs
Average over time: ∫F.dt / ∫dt = mΔv/Δt
If the force is not constant, they will generally produce different answers. Which to use?
We already have a well-defined concept of average acceleration: Δv/Δt. So for consistency we should define average force as mΔv/Δt. But we're not told Δt.
A reasonable punt at the average force here can be made by assuming SHM during the compression phase. This leads to 2mv2/(πs), i.e. (ΔE/Δs)(4/π).
A fully realistic model might be (imperfectly) elastic compression for some short distance, but with the force becoming capped at some point. E.g. there might be some hydraulic damping going on as fluid is forced out through membranes.
