Stick to Cressida Kroft, stay on her good side.
Ileosa does has any allies in the Vault, purchased for today; she could leave - or drop - the magnificent mansion and make the discussion more public... that's potentially a good idea, yeah, and she has some ideas for what she could say - "the Great Houses of Korvosa want to keep you down, even if your landlord's dead they think that you only deserve a tenant's life, they'll go against law and tradition and incite civil war just to spite you."
If that's her BATNA, she has something to threaten Kroft with! And anyone else with a moral compass - Lord Bromathan, maybe.
She only needs to split off one of the Great Houses, as they rule by consensus... what does she know about Lord Arkona?