First off Mink are extremely hard to shoot, kin it to trying to take a shot at a ferret, they can actually see the bullet coming and WILL avoid it.
get a hav-a-heart trap, you can bait it with one of the dead birds, tie it securely to the back of the pan area. once the mink is inside, a pellet gun will take care o the problem, put the muzzel in the cage , wait for the mink to bite the barrel and pull the trigger, quick painless and instant.
The only problem I am seeing with this situation is there were 3 of them meaning mother and 2 kits. you have to get over the cute little fur bearer syndrom and dispatch those babys also.
If you dont you are going to continue to have problems.
Brush up with a local search for preditors in your area. know what your looking at and for. It will help you in the long run.
Mink can get through amazingly small cracks in the walls, floors, wire etc, and they are not beyond pulling the wire off in order to get inside.
There is no such thing as 100% completely mink nor weasel proof, unless you are going to put them inside a total steel building.
those who dont think Otter leave the streams, I hate to correct you but they will in fact follow up tiny streams to find new breeding grounds, food, love etc. They will also leave the water and cross dry ground for many miles searching for another stream or river.
Mabe some also need to know their fur bearers as preditors.
I am in NH and live off a tiny stream and we do have the occasional otter here, in season the Trappers have taken them out of our stream.
Never assume you know everything there is to know, especially where as we all know the meaning of the word assume.