I know I answer a year later for you, but I was watching GOT too and I was looking for that exact same explanation ! If anyone else happens to look for it and winds up here :
This is what I was thinking with the context :
The guy's mother wants the boys to be "pretty" not for the King, but for the Queen. It yet goes without saying that the mother wants the Queen to be sexually attracted by them.
So "a sleek (=smooth) bit of mink (= a small, semiaquatic, stoatlike carnivore native to North America)" means "coquette", or "flirty woman". It's that exact "coquette" word that they use in the French subtitles.