Same or similar entry in other lexica; see the references in Photius kappa532 Theodoridis, and further below. The headword is the perfect middle/passive participle, masculine nominative singular, of the verb komyeu/w, "I make nice/quibble about/refine". This verb has in the middle/passive the meaning "be smart, be ingenious", referring to people (with a middle sense), and "made neatly" of things (with a passive sense) (LSJ s.v.). The former sense is exemplified by Philo Judaeus, De migratione Abrahami 75.6: oi( ta\ politika\ kekomyeume/noi, "those who are smart about politics". The latter is exemplified by Plato, Philebus 56C: kai/ tini prosagwgi/w| kekomyeume/nw|, "a certain ingeniously-made carpenter's square".
The use, as the headword, of a participle obviously suggests that these lexica are glossing a particular instance, and Plato's is the closest. The gloss e)kkallw/pistos is a hapax (Trapp, Lexikon der Byzantinischen Gräzität s.v., who renders it as as "ganz elegant").
Referring to people, the perfect participle is late (it occurs outside Philo in Philostratus, Life of Apollonius of Tyana 7.12.18, and in Procopius, Secret History 17.25.4). The start of the lexicographer's definition, however, is quoting Hesychius' entry for kekomyeume/nhs (kappa2086): "komyo/n means 'noble', or 'urbane'; hence the verb"; it is presumably referring to the same passage in Plato (the earliest extant instances of kekomyeume/nhs are in Gregory of Nazianzus and John Chrysostom). So there is no reason to think the lexicographer is distinguishing between "smart (of people)" and "elegant (of things)" in his definition.