Most likely you have a faulty master cylinder. If you just changed the brake pads and the pedal is still spongy, the air in the lines would have already come out with that job. What you're describing - the pedal going down a lot before the brakes actually engage - is typical when the master cylinder loses internal pressure, sometimes due to seal wear or a small leak you can't see with the naked eye.
Before you head to the shop, try this: with the engine off, pump the brake pedal several times to release system pressure, then start the engine and hold the pedal down for a few seconds. If the feel improves a bit once the engine is running, that's further confirmation the master cylinder is failing. Also check if there are any drops of brake fluid under the master cylinder or inside near the pedal - sometimes the leak is very small but enough to cause this.
It'd be unusual for something more complicated on a Corolla that age if it just started a week ago out of nowhere. The master cylinder is a repair that needs a shop, not something you can fix at home, but at least you know what to look for when you call.