The noun kēla that occurs in line is attested in Homeric and Hesiodic poetry by these three formulae at line initial position, line final position, and after the word break in the third foot:
#κῆλα Διός 'kēla of Zeus' Hesiod, Theogony 708 and Fragment 204.138
κῆλα θεοῖο# 'kēla of a/the god' Iliad 1.53=1.383
||πιφαυκόμενς τὰ ἂ κῆλα 'showing those kēla of his' Hymn to Apollo 444, Iliad 12.280
The formulas designate the weapons of Zeus in three places (Iliad 12.280 and the two Hesiodic passages) and the arrows of Apollo in three places (Homeric Hymn to Apollo 444 as well as here and in the repetition of this passage at Iliad 1.383). In the case of Zeus' kēla, it is evident that the reference is to his fiery thunderbolts. What is not immediately apparent is that Apollo's kēla are in fact also fire-bearing. Here is the line from the Homeric Hymn:
ἔνθ' ἄρ' ὅ γε φλόγα δαῖε πιφαυσκόμενος τὰ ἃ κῆλα
'there he was making fire blaze, revealing those kēla of his...
And in this passage, in the line that just precedes the occurrence of the formula shared by Zeus and Apollo, kēla theoio 'the god's kēla', we read of the fire of pyres as the immediate consequence of Apollo's arrow-shooting in the line immediately before the current one:
Iliad 1.52 βαλλ'· αἰεὶ δὶ πυραὶ νεκύων καίοντο θαμεῖαι
'he was hitting them; and immediately the pyres of corpses were burning thick and fast'
It is as though the kēla cause the pyres to burn. In seeking the etymology of this term, an underlying assumption has been, I believe, that what Zeus and Apollo share are arrows or things like them (thunderbolts), and so the etymology of kēla has been sought, unsuccessfully, in words in other Indo-European languages for arrows or 'shafts'. But every occurrence of the word in Epic diction evokes fire as an aspect of the kēla. What is happening on a thematic level is also relevant, because Apollo is here exercising mēnis 'cosmic anger' (see Iliad 1.75) by way of cosmic arrows — in fact he is functioning as a variant of Zeus, since striking victims with the thunderbolt is the conventional manifestation of Zeus' mēnis. And we will see that fire and in fact cosmic fire that goes up to the sky can become an expression of Achilles' mēnis as well (on this topic, see also now G. Nagy, "Cataclysm and Ecpyrosis, two symmetrical actions of Zeus as sky-god," Classical Inquiries, May 19, 2016, http://classical-inquiries.chs.harvard.edu/cataclysm-and-ecpyrosis-two-symmetrical-actions-of-zeus-as-sky-god/). On this level, one might prefer to define this word as 'fiery shafts' rather than just 'shafts'. It is not surprising that a connection was made in the Hesiodic scholia and in a late Byzantine Homeric gloss of this line which perhaps reflects earlier exegetical traditions:
Scholion on Theogony 708: <κῆλα Διός:> τὰ καυστικὰ σημεῖα "Zeus's kēla: burning signs/omens";
13th Century Byzantine on Iliad 1.51: ...ᾤχετο κῆλα θεοῖο. παρὰ τὸν κήσω μέλλοντα τὸν δηλοῦντα τὸ καύσω. ὅθεν καὶ τό· πυρὶ κηλέῳ, τουτέστι καυστικῷ, "kēla of the god: [derived] from the future [form] κήσω, which shows the [verbal form] καύσω. From where also the [phrase]: πυρὶ κηλέῳ 'with blazing fire' (Iliad 8.217, etc.), that is, καυστικῷ 'burning'") from Pseudo-Zonaras Lexicon, ed. E. Crusius 1808, rear. 1967, alphabetic letter kappa, p. 1206, line 3
It is reasonable to believe that the poetic tradition also sensed the relationship between κῆλα and the verb καίω, whether by way of folk etymology or, perhaps, by way of an inherited etymological relationship.
The adjective κηλέῳ 'blazing' is attested only in the formula πυρὶ κηλέῳ 'blazing fire', in synizesis, six times in line final position, once after the triphthemimeral caesura, and once in the form πυρὶ κηλείῳ before the penthehemimeral caesura. The form cited just above from the as the future of the verb καίω, namely κήσω, is not otherwise attested and may be a mistake, but the form ἔκηα (Iliad 1.40 etc.), which is an archaic athematic aorist with the 1st person singular ending -ṃ that was later misunderstood as a first aorist ending -α, attests to the verbal root for 'burn' with that vocalism (κη-). That form should be reconstructed as *ἔκηϝα, to which we can compare the Thessalian εκηυα attested in a ?? inscription.