We never say 'in the wink of an eye'. We say 'in the blink of an eye' probably because it is from a poem/play/story and was written that way. It is simple and direct and meant to be as such. "He drew his gun and shot John through the heart in the blink of an eye" The only reason we don't say 'in the blink of eyes' because it doesn't sound good and if somebody did say it they would sound like they didn't know English very well. When a saying is established we don't usually change it. A wink is whet we give to somebody when we are joking with them or with a third party that is present and we don't want the third party to know. A wink in the UK also means that you like somebody of the opposite sex.

Another question comes up to mind: why are indicator lights referred to as "blinkers" and not "winkers"? We usually use just one of them.

A wink is usually considered to be deliberate, to convey a message that something is a joke or there is some secret to be kept. Defined in this way, you would expect that it would be slower than a blink in order to make sure that the one you are winking at receives the message. Because of this, a phrase that would mean something very quick would probably tend to favor 'blink' over 'wink'. "In the blink of an eye" just developed that way. I'd say it rolls of the tongue a little more smoothly, but that is an opinion. There is no logical reason not to say "in the blink of eyes." ;-)