The only foolproof way of brake bleeding is by using a pressure bleeder.

The "two-man" method presupposes that both chambers are filled, the reservoir is never permitted to get low and the inside man never lets up on the brake pedal. Also for best results the brake pedal should not be fully depressed thus putting the master cylinder piston seal into potentially roughened territory.

The vacuum bleed method offers the potential of air being sucked into the system around the threads when the bleed screw is loosened.

A pressure bleeder will automatically keep the master cylinder full and air out. It also allows an easy one-man job with the ability to observe the state of the fluid while doing a complete flush. Pressure bleeders are reasonable in cost ($50 to $70).

BTW, I agree that you have air in your system. The only other explanation for a spongey pedal is too soft brake lines that expand when pressure is applied. Not likely is new lines were installed.