It is not terribly difficult for ImpSec to find the person who put her there - recent hire to the kitchen staff, competent but not overwhelmingly concerned with security when it was literally just a baby and not an explosive and he was offered a large chunk of change. Tracing back this chunk of change leads through a long meandering trail of people, variously shady, none of whom acted with particular malice beyond greed and the sort of vague dishonorableness of putting packages on the doorstep of the Imperial Residence.
The trail ends at someone who - under fast-penta - has no memory whatever of having interacted with the baby, the cash, the shady person who druggedly identified her, or anything else about the affair. She doesn't even have a similar-looking sister to explain the mixup. She has nothing in her background to explain how she would come by an offplanet baby or why she would convey it by such a circuitous method to the Imperial Residence. It's definitely not her baby - the lady's six months pregnant and the baby isn't that old.
In the meantime, the baby is parked with Droushnakovi, who found her in the first place; Cordelia, who was the first person she went to; and Gregor, as Drou is his personal bodyguard.
The baby and her basket are searched and found to be: a baby, and a basket. There is no note, it's the middle of summer and the child wasn't even in a blanket, the basket itself and the baby's onesie turn out to be local products that a Vorbarr Sultana thrift shop didn't even bother reporting stolen.
The baby babbles in her sleep. She's very cute.
"Hmm. Yes. The name is a problem," Cordelia agrees thoughtfully. "Perhaps I should do something about that."
"I am not at all sure 'they' will get any farther in tracing her origins than 'they' already have," says Cordelia. "The end they reached seems to have been pretty thoroughly dead."
"I'm not sure. It's... worrying, how everyone seems to be very focused on finding out where she came from and not at all focused on figuring out where she's going."
Drou makes an automatic check of the room to make sure that no one is waiting for her inattention in the rafters or behind the curtain, then goes to poke her head out the door and send the nearest servant to fetch Aral for his wife. The servant scurries and lets Aral know that "Lady Vorkosigan would like to see you, Lord Vorkosigan."
"It's this baby," she says. "I'm concerned for her future. I'm not sure I want to give her up to an orphanage."
"I think..." He shakes his head. "I think the course of action you are implicitly proposing may be more difficult than you realize."
"Well, I think this girl needs someone in her life who won't treat her like a potential explosive. Several someones. As many as will fit."
"Yes. I think it's a very elegant solution to the problems she faces. We're not going to let her potential as an explosive overshadow her potential as a human."