How did Kepler determine the orbital period of Mars? As I understand it, Kepler used the orbital period of Mars, along with observational data of Mars' and the sun's position in the sky to derive the orbits of Earth and Mars.  (As described, here: https://faculty.uca.edu/njaustin/PHYS1401/Laboratory/kepler.pdf)

But how did he determine the orbital period of Mars without already knowing the orbit of the Earth?
He knew the orbital period of Earth. You can do this by observing the sun's motion in the sky over a year. (And then you know that the orbital period of the earth is a year.)

Now from earth, you can observe Mars from one Opposition (https://en.wikipedia.org/wiki/Opposition_(planets)) to another. You can find out the time ($T_\text{relative}$ it takes for Mars to do this, using which, you find the relative angular velocity between the orbital motions of earth and Mars ($\omega_\text{relative}=2\pi/T_\text{relative}$). This angular velocity is the difference between the angular velocities of earth and mars. Using this and knowing the orbital period of earth, you can find the orbital period of mars.

$2\pi/T_\text{relative}=2\pi/T_\text{earth}-2\pi/T_\text{Mars}$

Hope this is clear.