The phrase "kick in" is usually used in context of something being activated in a short amount of time. Generally, it is also used when there is a period of time when there is no effect from the thing that "kicks in."
For example, let's look at "I'm waiting for the medicine to kick in."
- Notice that there is a period of inactivity between taking the medicine and the medicine's effect.
- The use of "kick in" implies that once the medicine begins affecting the body, its effects will be quick to materialize.
"Set in," on the other hand, is used for gradual motion/effect (rather than sudden), motion/effects that last a long time, and tends to be used in a less desirable context. (It's also most commonly used to describe undesirable weather).
As an example, look at: "We knew we were in it for the long haul when the snowstorm set in."
- Notice how "set in" in combination "long haul" makes the situation seem less than desirable.
- Using "set in" sets up an image in the reader's head of a slow-moving mass of clouds that moves towards the speaker, and, once it reaches the speaker, lets loose a snowstorm that lasts for several days.