What sentence between those two is valid, using singular or plural with "any kind of" ?
Any kind of car
Any kind of cars
my search in "google ngram" about your question showed me that all is right with both of these sentences. But for me they have slightly different meaning from each other.
When you talk about car from range of them, you can use "Any kind of car" and anyway it will be talking inside category of usual cars (if you don't have any specific context with your sentence that can change meaning).
When you talk "Any kind of cars" it can be about different more common categories of different cars (and might suppose attention on these categories in your talking).
1) About any car from one category of cars (like usual cars not specific).
2) About any category from different categories of cars (like sport cars, family cars or so on).