----"It must be something very simple and I'm just too blind to see the obvious but I just can't see Irelia winning this matchup."
Don't worry. I don't see it either.
Irelia isn't the best counter versus Jayce. I highly recommend Yorick. He's actually extremely powerful top lane right now, as you might have noticed on the forums when people have asked, "How do I counter __ top?" - "Yorick. 'nuff said." Other good counters I have found from personal experience include
Why? Because they have a lot more sustain than jayce, and can set up very good ganks with a jungler because of their kits. In addition, they have nice harass while Jayce's harass is skill-shot based, so it is avoidable.
And finally, Viktor. If you haven't read Jayce's lore, read it. WHO DARES QUESTION SCIENCE
----But I want to play as Irelia!
And so here are some tips on how to 'counter' Jayce, because otherwise this would be a pretty off topic answer.
First of all, make sure you have a point in all of your abilities. I tend to go Equilibrium Strike first, in case I get into a fight early or if their jungler ganks before I hit level 2, I can easily stun and get out. Then, I get Bladesurge at level 2, getting Hiten Style and levels 3, 4, and 5 to maximize my healing.
- Try to play safe and passively until about level 5. Wait for Jayce to get your hp down just a little bit lower than his, then BLADESURGE ONTO HIS FACE, hit Equilibrium Strike, and start going at him with your Hiten Style. The reason I suggest level 5 is because Jayce is ranged, and unfortunately, you're not. If you Bladesurge onto him at earlier levels, minions can hurt. A lot.
----Note Know how much damage your Bladesurge will do to him. It does 20/50/80 +1.0 Attack Damage, so make sure after you Bladesurge that his health won't be lower than yours, or else you'll be left with a measly slow or be forced to wait until he gets his health higher.
- Remember how I suggested some champions to counter Jayce because they could set up good ganks with a jungler? Well, Irelia can do that too with Equilibrium Strike. Again, wait for your health to be just a bit lower than Jayce's, stun him, and work together with your jungler to pick up the he kill.
----Note Usually, especially in higher levels of play, the jungler is supposed to make the first move in order not to arouse too much suspicion. However, if you can, make the first move by using Equilibrium Strike to stun Jayce first, so that the jungler has time to close the gap in between the river and wherever Jayce is. However, it's still situational. If you and Jayce are both at low health, it might be best for the jungler to go in first.
Don't push too hard. Irelia is great at getting high creep score with her Bladesurge as well as the healing and extra true damage from Hiten Style. I find that her kit makes it easy to get CS under the tower. Note that Irelia can push fairly hard, so make sure you're not overextending, making you prone to jungler ganks. You're a melee champion, and Jayce can safely kite you at his tower until later levels.
Buy what counters Jayce and the rest of his team; Do not always follow exactly what the LolPro build says. It's a reference, kind of like a note card when you're giving a speech.
Dodge his shock blast, especially when fired through acceleration gate. The 40% increased damage hurts a lot, and could constantly send you back to base, putting you behind.
- Regrowth Pendant can be good for recovering after being hit by one of those bad boys, but I personally like to run Boots of Speed and Health Potions. Why? Because it helps me dodge Shock Blast. It's kind of common sense, but, when Jayce puts an acceleration gate right in front of your face, he's probably going to shoot a Shock Blast through it so try to predict him, and dodge it, or even Bladesurge a minion to avoid it.
I've compiled Jayce and Irelia's stats and compared them to see who scales better. The better stat at level 5 is in bold.
Jayce vs. Irelia
- Health 420 (+90) ---- 456 (+90)
- Attack Damage 46 (+3.5) ---- 56 (+3.3)
- Health Regen 6 (+0.8) ---- 7.5 (+.65)
- Attack Speed .658(+3.0%) ---- .665(+3.2%)
- Mana 240 (+40) ---- 230 (+35)
- Armor 12.5 (+3.5) ---- 15 (+3.75)
- Mana Regen 7 (+.7) ---- 7 (+.65)
- Magic Resist 30 ---- 30 (+1.25)
Good luck beating Jayce with Irelia! :)
---- Credibility of this Answer - League of Legends player for two years, and I own both Jayce and Irelia with experience with both champions, more notably Irelia. I have also won top lane using the alternative champions I suggested at the very top.
Champion stats found on leagueoflegends.wikia.com