The sound seems like that's the brake light relay clicking on/off. At the same time it seems like you loose power brakes based on your description of the hydraulic feeling.

Maybe ABS control module. See this description of how the ABS system works http://auto.howstuffworks.com/auto-parts/brakes/brake-types/anti-lock-brake1.htm loosing that hydraulic feeling seems like the control module is isolating all brakes in blocking mode.

This may be one of those repairs to take to a specialist.