Irelia vs Kayle Matchup (Please Help)

Who is supposed to win this match up? Because I played it, and it seemed easy at first but boy did Kayle have a surprise for me. I was up 1 kill, had the cs lead, had the level lead, and had the item lead. I had Bilgewater Cutlass and Recurve Bow with Doran's Shield and 1 potion and boots. Before anyone asks "why didn't you go corrupting potion?" because I was told to go Doran's Shield and 1 potion if you weren't confident in the match up. Alright now for Kayle: She had 1 kill, and had Berserker's Greaves and Doran's Ring. I was FULL HEALTH and she had like 1/5 missing. And no it wasn't minion aggro. I went in for the kill (with all my stacks) and got completely smashed even though I was higher level, had higher cs, and had more items. Usually when you're against an Irelia with 2 kills and has higher everything than you, you die. But no no no, this was not the case. And no she did not use Ult.

So can someone who maybe is familiar with this match up or maybe familiar with Kayle and her stats tell me why she won that fight. Because I don't get it. When I look it up, it says Irelia is supposed to win, and I was until this miraculous power spike.

And before I press "Post Discussion," I have to add that it gets better. Now, after her second kill, she can literally just walk up to me and kill me. Minion aggro does nothing to her no matter how big my wave is. I can't run from her because she Qs me. I can't fight her because she kills me too fast. After that third kill, even if I get her low enough she just ults. And if I just try to farm under turret, her E just fucks me. Then she can literally just turret dive me. Its like I'm forced out of lane, but if I go to other lanes, she clears minions and shreds towers ridiculously fast.