Firstly,
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×Are you placing the mimic somewhere where he isn't going to go wandering off while you get to where the eyes can see you?
i.e.
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×in the box (or between the orbs above it, I guess)
If that's all ok, then the easiest way is to
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×make sure that you're directly beneath the mimic. Down-eyes should be easy pickings - for the diagonal ones you just need to make sure your sword is in the right direction, get their attention and you should be able to reach them in time by moving across.