They don't have very much to work with, but they don't have nothing either.
There's been one death by heart attack every twenty-four hours like clockwork. Maybe more than that -- there have been a handful of cases of inmates getting heart attacks at other times -- but every other case has had some sort of pre-existing risk factor, like "being eighty-seven years old." Their current working theory is that the deaths happening every twenty-four hours are all connected, and other deaths aren't.
The "untraceable poison" that her father mentioned at dinner is the current public theory. She doesn't think it's right, and neither do most of her colleagues. Poisons don't work like that; you can't time deaths that precisely. More to the point, implementing it would be nigh-impossible. Some of the deaths are halfway across the world from each other, and even if the killer is flying from Sydney to Los Angeles to England to Vancouver to Japan (they aren't, unless they're travelling under several false identities) they'd still need to somehow administer the poison.
That admittedly doesn't leave them with much in the way of plausible methods.
Even so, that's not the only information they have. The biggest clue is the countries. Several Anglophone countries, a handful of deaths from non-Anglophone countries that were reported in English, and a weirdly high concentration of deaths in Japan that were only ever reported in Japanese. The current working theory is "an American, probably living on the West Coast based on the timing, who speaks Japanese," but they aren't ruling out "someone from some other Anglophone country who speaks Japanese" or "someone from Japan who speaks English" or for that matter "someone from somewhere else entirely, who speaks English and Japanese and has carefully been avoiding killing people from their actual country."
They think it's probably not the last one, but that's a guess about psychology more than anything.
The point is, the killer speaks English and Japanese, which gives them strictly fewer possible suspects than if they'd stuck to one or the other.
(Theoretically it could be a conspiracy. It technically makes the poison conspiracy more plausible. She doubts it, though; it would be weird for a conspiracy to target people whose deaths were reported in exactly two languages, and not bother to pick up anyone who knows Mandarin or Hindi or Spanish or any other widely-spoken language.)
Then there's information about the victims themselves. The most obvious clue, the one they picked up on as soon as they knew there was anything to pick up on, was that all the victims had been accused of serious crimes, primarily murder, and all of them had been let off easy, either by being acquitted or by receiving something lighter than a death sentence. There are other patterns, if you look for them -- disproportionately cases where the original victim was a child, disproportionately cases where the accused was let off on a technicality -- but they don't have a large enough sample size to be certain, yet.
Most of the people dying had mugshots published in the same articles that talked about the cases in the first place. That's not really surprising; most plausible murder methods require you to know who your victim is.
The weirder part, the part she didn't entirely believe her colleagues about until she double-checked the analysis of the agent proposing it, is that of the people whose mugshots weren't published, all of them had very uncommon names. That, she can't explain.