The most persuasive discussion of the etymology of "on the blink" is this article that investigates the etymology of the American expression "on the fritz," for which it notes that the British and Australian equivalent is "on the blink."
A key part of the article quotes yet another article, which notes:
The phrase is now a common expression meaning that some mechanism is
malfunctioning or broken. However, when it first appeared — about 1902
— it meant that something was in a bad way or bad condition. Early
recorded examples refer to the poor state of some domestic affairs,
the lack of success of a stage show, and an injured leg — not a
machine or device in sight.
It goes on to observe:
Some people have suggested it might be an imitation of the pfzt
noise that a faulty connection in an electrical machine might make, or
the sound of a fuse blowing. This theory falls down because none of
the early examples is connected with electrical devices, and the
phrase pre-dates widespread use of electricity anyway.
The article finally reaches the conclusion:
I’ve gone around the houses, considered this theory and that, but come
to no very definite conclusion. But the truth is that nobody really
knows, nor now is ever likely to.
I would suggest that the expression might be connected with the fact that, when we blink, we stop seeing -- our vision effectively stops working. Something that's on the blink would then be in a state where it's not working, or not working fully.