How to find the period of a pendulum knowing only its speed and maximum arc-length?

The pendulum has a velocity of zero exactly twice per period - once at either end of the swing. A very simple approach to find the period is to measure the time between velocity minima, and double it.

Speed at a single point in time won't allow you to calculate the period. Imagine a pendulum of fixed length, which is released from different (but small) heights. The speed of the pendulum at its nadir depends on its angle of release, but its period does not. Since the same period can be achieved with many different nadir speeds, you can't calculate period from just nadir speed.