You don't start with a departure date--that's one of the answers, not one of the inputs.
I don't know a formula for calculating a gravity assist orbit so I will use the easy case and omit Venus, as well as assuming the planets are in circular, coplanar orbits:
Find the orbital period of the transfer orbit. For our simple case the orbital radius is (mercury ...
Since I had nearly the same problem, here is what worked for me:
Get two vector ephemerides from Horizons, one for Voyager and one for Jupiter. Choose the Sun as the Coordinate origin.
Subtract the VX, VY and VZ values of Jupiter from the corresponding values of the spacecraft to get a velocity relative to Jupiter.
Calculate the angle between that vector ...