As Queen of Korvosa, Ileosa is of course still in mourning. The performance of grief is tedious, and she has to keep it up for at least a week by the rules of Korvosan public opinion. But in private she can use her more determined, purposeful cast to her face.
Remember what she stands to lose. Power, freedom, eternal life and all its pleasures. Almost everyone would gladly see her deposed as queen of Korvosa, her reign hangs by a thread. If she can bind this little party together, use them as exemplars for everyone who wants Korvosa to have a strong ruler, they could be a lifeline for her.
(They'll need a name. The Silver Maidens? Imagine them all outfitted in shining armor and shimmering cloaks, lined up neatly like dolls, her personal knights devoted to serving her alone - focus.)
Oriana and Tisharue are blunt instruments, Vavana a kindred spirit, Caron shattered and ready to be reshaped should she hold herself together, but Weiss...
She sees Ileosa as a monarch instead of a person. Normally that would be no cause for complaint. Desirable in a subject. But from her words and expressions Weiss seems to invert the importance of the two, as though monarchs are the replaceable hollow objects to pay lip service to, and ordinary people the ones worthy of respect and service!
To secure control over her, Ileosa will need to do something drastic. Either indebt her with protection and payments to her family, wherever in Varisia they are, or somehow convince her to see her as both a monarch and a person.
For the former, she'll have to set up the right hints in advance, wait and see whether her agents succeed. For the latter, shows of emotion, less of the reserve, in the faint hope that Weiss' indiscriminate heartstrings will tug her to save her queen as readily as she would some random guard or petty noble. There is one more strongly persuasive option that comes to mind, but that will also require patience for it to grow to fruition.
In the meantime, chasing down a cold case should keep her occupied. If the little fox does sniff out that serial killer, that could play into her plans as well.
Remember, Ileosa, not your people but the people, praise her service to the city not the crown. Ugh, talking Good is so painful to adjust to.