Shutting your eyes makes the mirrored caster invisible to you.

In order to discern the square of an invisible creature you may attempt a Listen check at a DC equal to the creature's Move Silently check +20. If you succeed, you suffer the normal 50% miss chance due to total concealment.

Mirror images don't produce sound, so you cannot detect them and they cannot foil your listen check. If they occupy the same square as the caster, however, there is a gray zone.
My opinion is that you can't be foiled by vision, so if you reach the AC and beat the 50% miss chance you'll strike the real caster.

No, mirror images aren't enemy creatures

Sneak attack requires "another enemy". Mirror images aren't enemies different from you; they're in your space, and they "move with you", so they don't present additional threats.

In general, there's no strict definition of "enemy" provided in the PHB, but it's often used interchangeably with "hostile creature". So, as you continue to find ways to enable Sneak Attack for your Arcane Trickster, consider the Sneak Attack rule to be:

You don’t need advantage on the attack roll if another creature hostile to the target is within 5 feet of it, that creature isn’t incapacitated, and you don’t have disadvantage on the attack roll.

Mirror images aren't hostile creatures, so this won't work. RAI for allowing sneak attacks without advantage when an enemy is adjacent to the target is clearly that sneak attacks are facilitated by the rogue supplementing other party members in combat, so low-level illusions don't cut it. Note that you don't have to be in melee; you can stand back and snipe with missile weapons while your ally gets in close.

(As a general suggestion, I would work on using your Arcane Trickster's spells to buff yourself/teammates and make ranged sneak attacks rather than trying to make melee sneak attacks work without teammates to help you).