Welcome to world building.
Actually it’s quite a simple problem if the planet has a slow rotation rate. If it takes 14 Earth days to make one complete revolution then it will have one week “days” during the summer and a one week “nights” during the winter. Incidentally it will also have one week “nights” in the summer and one week “days” during the winter, so its “days” and “nights” are always one week long.
To have the same effect across the whole planet it would be necessary to have a 0 degree axial tilt rather than the 23 degrees Earth has. Highly eccentric orbits might cause some disturbance due to motion of the planet covering a significant distance in one week putting the timings out. A circular orbit wouldn’t have this problem but wouldn’t have seasons either.
If the intention was for 1 Earth day “days” and 1 Earth week “nights” in the winter and the reverse in the summer then unfortunately it is not possible because a planet can’t be speeded up or slowed down as quickly as you need due to the amount of rotational energy involved and the law of conservation of angular momentum.
Assuming that the effect is needed to be planet wide and uniform. If not then the situation arises on Earth already.