Lady Kirei gives Iri to Gil first, so that Iri leaves this world unfaithful to her husband. She then taunts Iri about how she's going to raise Ilya as a second daughter.
After the War, Lady Kirei does everything she can to convince the Einzberns to let her raise Ilya as a ward and "prepare" her for the next war. Which, as one of the few surviving Masters and the next referee, she might just pull off. Lady Kirei gets Ilya to call Kiritsugu on every anniversary of Heaven's Feel and reiterate how lonely and disappointed she is that Kerry never came back for her, "betrayed" the Einzberns, etc.
...And how, despite Kerry's "betrayal", Ilya will become a "Hero of Justice" like her father and fight the masters in the next war. Because for some reason, I think Kiritsugu revealing his "Hero of Justice" dream would piss off Lady Kirei even more than it did Canon Kirei.
Lady Kirei's new ward would give Caren Ortensia additional abandonment issues, too, so that's also nice.