Booster

Sounds like you have a leaking power brake booster. You still should have brakes if you push really hard on the pedal.

Not what? No brakes? Does the pedal go to the floor? Is it just real hard to push? What changes at 10 mph?

I'm still not clear on what the brakes are doing. The pedal goes down slowly, so are you saying it normally goes down faster or it's going down too far and the car doesn't stop?

If the pedal is going down too far as you hold steady pressure on it, there could be a leak in the hydraulic system. If the brake fluid is low in the reservoir, check underneath for signs of leakage, particularly inside each wheel and along the left side of the car. If the fluid level is good, there could be internal leakage inside the master cylinder.

Another thing you can try is to stop the engine, pump the brake pedal a few times, then see how hard it is to push. With the power booster exhausted of vacuum, the brake pedal will push very hard but the brakes will still stop the car. That will give you an idea of what they will feel like if the booster has failed.