Yeah, me too. Don't the Klingons search their captives? I mean, they're known for concealing weapons in their uniforms, so divesting prisoners of their clothing should be standard practice. And if that patch emits a strong enough signature to be detected from parsecs away, wouldn't the Klingons have been able to detect it too?
Can't recall where I read it, but someone explained that said patch was of a color that Klingon eyes couldn't detect. Maybe that was mentioned in some version of the script.
The Klingons have sensor technology. Why would they rely on eyes alone? For that matter, the core question remains, why leave prisoners in their own clothes at all? In our prisons, prisoners have to give up all their clothes and possessions and wear prison issue. That's specifically to prevent them from sneaking in contraband or weapons. And as I said, we know from "Heart of Glory" that Klingons build weapons into their uniforms. It makes no sense by any standard that the Klingons would've let Kirk keep the same uniform he'd been arrested in, without at least scanning it first.
Anyway, it sounds like that "someone" was cribbing a concept from the novel Pawns and Symbols by Majliss Larson, which claimed that Klingons had different color pigments in their eyes and thus couldn't see the color red. If that were true, then movie-era uniforms would appear the same black as the patch. But that claim about Klingon vision has never been made anywhere else besides that book.
As Vulcan has no moon, who knows how long "three months" might be.
The line was Kirk's, so it stands to reason that he was using standard (Earth) months. Heck, everyone in ST seems to use Earth time units regardless of their planet of origin.