It sounds like there is air trapped in the system. Check that the rear chamber of the master cylinder is full of brake fluid, then bleed all the brake calipers again. It's best to use a pressure bleeder. The sequence is right rear, left rear, right front, left front.

As you mentioned it's best not to press the brake pedal to the floor when using the partner/pedal method. Typically you place a spacer (say, a piece of wood) under the pedal to prevent it from going to the floor. If the calipers are leaking you will see brake fluid dripping from the calipers.

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Fred Hoelzle