Havent played this matchup yet but here's some points I could make:
-Jax isnt very meta in diamond+, he was in a decent spot when Irelia was op last season, but now there's no reason to ever pick jax into ANY matchup if you really want to win
-The best top laners right now are tanks and champions with sustain: Mao, Naut, Darius, Poppy, Illaoi, Renekton, Nasus, Yorick
-Jax is in a bad spot. The problem is his kit is so basic that they cant give him any buffs because he would stomp low elo really hard due to being an auto attacking champion rather than a more mechanical champ
-Illaoi lands one e on Jax and he's pretty much done for until triforce, bork, armor item like randuins, and even then I'm not sure if he could take her
-Illaoi q= infinite sustain jax=no sustain at all in his whole kit, most champions can poke him out of lane and get a really good early farm advantage due to having more sustain
-Jax has almost no winning matchups right now unless you're straight up a better player than the enemy top. If both players are pretty equal, Illaoi should win almost everytime. Im pretty sure at level 6, even if jax gets a kill on her, she can still take him 1v1 with ult
-yes jax can dodge her w with his counter strike, but if he gets into that range it gives her a free opportunity to land an e, if she lands that she can then w that and doesnt even have to hit and can just hit your vessel so your dodge doesnt matter
tldr:dont pick jax until he gets changed, better matchups into illaoi are darius, gnar, fiora, singed, cass top. Even in these matchups illaoi ult can still one hit u