How do you split push against illaoi late game with a melee champion?

if your champ > illaoi late game: kill him if your champ < illaoi late game: split the opposite lane he is in.

Then you're doing it wrong, split push is so that you don't have to go against enemies. Ward the fog of war, push lanes and back off when you think you did good enough or continue pushing if the whole enemy is on your map in mid lane.

? Don't get in her melee range. Or dodge her E. You really shouldn't be 1v1ing her and be expecting to win, outside of being some of the champions that counter her. It's the same idea as a melee champion trying to 1v1 a Tryndamere splitpushing. He will probably kill you because he's a lot better at dueling unless you have CC or an AS slow or are an equally good/better duelist.

Kill the creeps before they can push to tower long enough for said enemy champion to chunk it. And said splitpusher will usually be the top or jungler, meaning them being gone is better for your team generally, as long as they can handle not having you there. It's the same treatment as any other fast splitpusher. Ward ahead to see them trying to push earlier, have at least one person on the team get a Zz'rot's, etc.