When I press the brake pedal at high speeds, the car shakes.
It sounds like you have warped brake rotors. Your vehicle has rotors on the front and rear axles. The rotors can cause a vibration when they are not straight, or have thickness variation. If the steering wheel shakes, the front rotors are most likely at fault. You have two choices to repair this -- a shop can machine the rotors, or they can be replaced.
The brake pads will be inspected while the brakes are apart. If they are have less than 25% left, then this is a good time to replace them. If the rotors are heavily rusted or too thin, they will need replaced. If you decide you want your rotors replaced, consider having a certified mechanic, such as one from YourMechanic, come to your home or office and perform the service there.
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