This doesn't feel like any air problem I've ever had, where the brake pedal is soft, maybe to the floor, but if you pump the pedal a few times it will compress the air enough so that the pedal gets hard and stays hard as long as you've got pressure on it. Release the pressure, the pedal goes soft again.
In this case, the pedal is hard at first, then goes to the floor with steady pressure, and doesn't pump up if you keep working the pedal. This car doesn't have ABS brakes. Is there a some other gizmo that could be bypassing here?
As far as bleeding goes, would it work to run a tube all the way from a brake caliper (starting at the diagonally furthest away) back into the reservoir, and bleed the system by pumping the brakes until bubbles stop coming off. .