"It can not doesn't work" is incorrect. 'Can' and 'do' are two completely different statements about a verb. "It can work" means that it is possible for 'it' to work. ie. It is able to work. "It does work" means the verb ('work') frequently happens. Most of the time, 'do' is omitted unless the sentence is negated or if it's a question (ie. "It does work" = "It works"). "It cannot work" means that 'it' is unable to work. "It doesn't work" means that it isn't working. (ie. The statement "It does work"/"It works" is false). If you want do say that it doesn't work and that it cannot work, you can say "It can not and does not work" (or "It can't and doesn't work").

I just noticed the additional details you added. If you want to use a double negative, you can say "It cannot not work" ("It can't not work") ie. it not working is impossible.

It doesn't work is correct. It means that it might has some problem working.