This is the simplest and cleanest Kayle reroll guide you can find online. Let's dive right in!
There are two main variations:
1. Elderwood Solar Kayle

This version has a higher cap, especially if you have certain Ornn Artifacts on Kayle.
Kayle Artifacts: Flickerblades > Wit's End > Statikk Shiv > Seeker's Armguard > Gambler's Blade > Mittens > Zhonya's Paradox
Only play this version when you have Prismatic Ticket or Hedge Fund. Otherwise, you won't have enough econ.
Cursed Crown is the 2nd best augment you can pick for this comp.
Because it will allow you to play 8 units at Lv6.
Both Hecarim and Elise are fine as your last champion to 3★.
Before Kayle is 3★, Xayah is much better at carrying items.
Itemization
Kayle's items can be flexible. Only Guinsoo's Rageblade is core.
BIS: Guinsoo's Rageblade + Jeweled Gauntlet + Hextech Gunblade
Flex items: Rabadon's Deathcap, Giant Slayer, Red Buff, Striker's Flail
When to Roll / How to Roll
Mostly at Level 5, you can roll down to 20 gold aggressively if you are close to Ornn or LeBlanc 3★. Both champions can bring extra resources to your board.
Try to 3★ Ornn and LeBlanc first if you have to choose. Ornn's quest gives you an Artifact, and 3★ Ornn doubles the processing speed. Meanwhile, LeBlanc's passive gives you a copy that can be sold for econ.
2. Inferno Solar Kayle

This version is much easier to play and much stronger in the early and mid game!
Pick a strong early-game augment to seize the tempo! Slam items and roll for 2★ upgrades! Make the most out of the tempo of a reroll comp and crush your enemies in the early/mid game!
"Portable Forge" with good Artifacts on Kayle is the strongest early option. We mentioned which Artifacts are good on Kayle in the last chapter.
You can also play this version when "Worth the Wait" gives you any champion in this comp, since you will have very strong tempo with an early 3★ champion.
Later on, you can add Lunar units such as Aphelios and Alune for a higher cap. However, you should generally prioritize seizing early tempo over seeking a higher cap.
Positioning
Leona's passive gives her extra Armor/Magic Resist during the first 10 seconds, so make sure she takes damage first.
Itemization
Since you have Inferno for Burn and Wound, and Kayle has magic penetration built in, you don't have to build utility items such as Sunfire Cape, Ionic Spark, Void Staff, or Morellonomicon in this comp.
The only utility item you might need is Last Whisper on Varus, but it is very low priority.