I realize it is probable that the use of this name goes back to Anouk Aimée. So before we assign either a Russian or an Inuit origin to it, shouldn't we know how Ms. Aimée came to have the first name of Anouk? Anouk Aimée is evidently a stage name, created around 1947 when this actress first began appearing in films. Do you have evidence from a biography of Ms. Aimée as to how she created the name?
I have no doubt that you have been told by someone that Anouk has an Inuit origin, but that isn't enough evidence for this site to list it as such. You would have to provide references from a dictionary of an Inuit language, or at least evidence that Anouk was a name in use in Greenland, Canada, or Alaska before 1947 to convince scholars that this was a correct origin.