Play "DNA Surgery" and choose Machine. Flip this card from face down/summon it and activate "Machine Duplication". While you won't draw from all your Dark Mimics, you will surely be able to level a few of them up, even if you don't stop your opponent from attacking. "The Dark Door" can make sure only one of your monsters dies a turn.
Use "Waboku" when this card is face-down on your side of the field and about to be attacked. You'll draw a card, it will replace itself with "Dark Mimic LV3" on your next standby phase, and your opponent's monster will still be in a position to destroy the LV3 by battle the next turn, granting you even more cards. Properly used, the two Dark Mimic cards can have a massive deck-thinning effect.
Even better, use "Broken Blocker" and, provided your opponent has only one monster to attack with (use "The Dark Door" if otherwise) you will end up with two Dark Mimics ready to level up and while you won't get the draw from them, you will draw 4 from the LV3s and will have already thinned your deck by 9 (1 drawn from "Dark Mimic LV1", 2 "Dark Mimic LV1s" summoned by "Broken Blocker", 2 "Dark Mimic LV3s" when the "LV1s" level up and 4 cards drawn when the numerous "Dark Mimic LV3" are destroyed).