Sometimes, the very thought of something makes you sick; you are horrified or repulsed by it. That’s what the expression `sicken at’ means.
*The children sickened at the idea of having to clean public toilets.
When you `sicken of’ something, you get bored or tired of it. In other words, you get sick of it. It is also possible to sicken of a person.
*Many students were beginning to sicken of the strike.