Plasma sources with controlled and reproducible electron densities over metre-scale lengths are essential for extending the acceleration length in plasma-wakefield acceleration (PWFA), while ensuring stable wakefield formation. In the AWAKE experiment, this motivates the development of scalable plasma sources with highly uniform electron density profiles, together with diagnostics capable of characterising them. We report on a laser interferometry diagnostic developed to measure the axially averaged electron density in a 10 m discharge plasma source for AWAKE. We introduce a practical analysis method enabling reliable, time-resolved reconstruction of the plasma density, adapted to the fringe distortions introduced by the long optical path. Reproducible axially averaged electron densities with peak values from approximately cm to cm are obtained for argon discharges at 24 Pa with peak currents between 150 A and 500 A (current densities of order A cm) and pulse durations of tens of microseconds. This diagnostic provides a robust tool for characterising metre-scale plasmas for AWAKE, as well as for other PWFA experiments.