"He do have eyes. "
When I first read Blackest Night: Superman #2, I wasn't expecting Black Lantern Psycho-Pirate to have his head fully restructured, but having seen other Black Lanterns be completely torn asunder only to be fully reconstructed by The Black Lantern Power Rings in mere moments I guess it's only natural that he would have been whole upon being "resurrected." Ah, that reminds me, as it has been revealed, a Black Lantern isn't actually a person who has been resurrected from the grave, they are actually The Black Lantern Power Rings.
See, when the rings attach themselves to a corpse they don't bring back the person who has died, instead they take total control of the corpse and begin to operate it like a machine. Also, they use the memory engrams imprinted on the corpse in question's mind to impersonate the person whose corpse they are in control of almost flawlessly, sort of like how Noveno Espada Aaroniero Arruruerie was able to impersonate Kaien Shiba to the point that he actually convinced Rukia Kuchiki that he was indeed Kaien Shiba...at least until he asked her to bring the heads of Ichigo, Renji, Uryu and Chad to him.
This allows The Black Lantern Power Ring to maintain the illusion that everyone it comes into contact with will be under the impression that they are indeed who they say they are. So...now the question we have to ask is can Aizen use Kyōka Suigetsu's power against a ring? And even if he can...how is he supposed to kill Black Lantern Psycho-Pirate without both a Green Lantern Power Ring and a Power Ring from any other portion of The Emotional Spectrum? So far nothing other than reinforcing any single one of the six other lights (i.e., The Indigo Light of Compassion) with The Green Light of Willpower has succeeded in permanently destroying a Black Lantern.
Well, actually there could be an alternative solution, we don't know what effects high-level Kido spells or spiritual energy (in general) that is as strong as Aizen's (or maybe ever weaker than Aizen's) would have on a Black Lantern. Perhaps...if he can keep himself from being affected by Black Lantern Psycho-Pirate's ability to impose his emotions on other beings (which would be hard as Black Lantern Psycho-Pirate can impose his emotions on others even if they are not looking directly at him) Aizen could crush The Black Lantern Power Ring (which are incapable of reforming themselves after being physically obliterated) that is operating Black Lantern Psycho-Pirate's body using his monstrous spiritual pressure, thus resulting in Aizen's ultimate victory over Black Lantern Psycho-Pirate.
And, since Black Lantern Psycho-Pirate's body would be disintegrated by the small but semi-potent explosion unleashed by The Black Lantern Power Ring's destruction, Aizen does not have to worry about another Power Ring coming from out of the blue and rebuilding Psycho-Pirate's corpse. This is all just theory and speculation of course. For all we know, The Black Lantern Power Ring operating Black Lantern Psycho-Pirate's body might just end up absorbing all of the spiritual energy around it, therefore reducing Aizen into a powerless weakling who would be unable to defend himself against Black Lantern Psycho-Pirate and would then have his heart ripped out which in turn would result in the "birth" of Black Lantern Sōsuke Aizen.