You have simple harmonic motion. The damping term (proportional to dx/dt) only affects the amplitude.
Look at the period. The peak is roughly at 0.5, and again at 5, leading to a period of 5-0.5=4.5.
The period of simple harmonic motion is 2*pi*sqrt(m/k). You know m=20 from the given. So:
4.5 = 2*pi * sqrt(m/k), solve for k gives close to 40.