If you're sure the linkage is connected, I really think you need to bleed the brakes. Either using a pressure bleeder, or the old pump-it-up & hold-it-down method.
With pressure in the bleeder or the pedal pumped & held to the floor, you should get fluid out of the bleeders. As the fluid goes down in the master cylinder, refill it before it goes empty. Once you've gone through some fluid, air should start coming out too. Anytime air has come out, you should get a little more "pedal".
Even with a really bad master cylinder, you ought to get some results.
You don't need air pressure to bleed the brakes.
Eric
CBVET