"Ingrid," I say at one point, "can we just leave the emotion out of it for a moment and concentrate on the figures?" "We can," she says. "But it's so hard. It's hard if you care about animals." She might be one of the most infuriating subjects I've ever interviewed, but no one could ever say of Newkirk that she doesn't care.
She's devoted almost her entire adult life to defending the rights of animals and even though she's now 63, when I ask her if she has any plans to retire it's as if I'd asked if she fancied a nice veal cutlet.
"No! Never. Peta is my life. It's everything to me."
She's up before dawn, answering many of the 800 to 1000 emails she gets each day personally, and although the encounter with Wintour was 20 years ago, it seems emblematic: they're both formidable Englishwomen of a certain age.
They've both imprinted their personalities on the American institutions they've led. And they both acutely understand the power of the visual image.
Newkirk, however, is by far the less well known. Yet there's no denying the impact she's had on the world of animal rights, animal testing, factory farming and fur wearing, not to mention celebrity endorsements, and what can only be described as the use of shock and awe in advertisements.
It's because of Peta (People for the Ethical Treatment of Animals) that we're faced with an almost daily onslaught of celebrities taking their clothes off for some cause or another.
Peta started the craze way back in 1994 when it photographed five supermodels, including Naomi Campbell, minus their clothes, with the caption: "We'd rather go naked than wear fur." Campbell, of course, being the principled idealist that she is, went on to wear fur about two seconds later. "I had to sack her," says Newkirk.
"We let it go the first two times because she said it was a mistake, but the third time, I said 'just sack her'." Still, the campaign has run and run.
"And everyone has followed our example. That movie Calendar Girls would never have been made if we hadn't done what we'd done. Everyone does it now, but at the time it was just a silly gimmick we came up with." A silly gimmick that propelled Peta on to the pages of every broadsheet, tabloid and celebrity magazine in the world, and which it has never left. It was one of the first non-profits to really "get" celebrity, and that is still a major plank of its strategy.
Its latest film shows Joaquin Phoenix drowning, an image designed to draw attention to the plight suffered by fish. Peta tried to buy air-time during the ad breaks in the Oscars to show the film, but the network banned it.
The film represents Peta's signature cocktail of graphic imagery, celebrity firepower and sniff of controversy - though it is pretty mild stuff by its past standards, which include poster campaigns comparing people who eat chickens to Nazis, factory farming to the Holocaust, the Westminster Dog Show (the American version of Crufts) to the Ku Klux Klan, and dog breeders to slave traders.
Subtle it is not.
Is it Peta's strategy to upset everyone, I ask Newkirk. "No," she says. "Our mission is to provoke thought. People have been taught to disregard what happens to pigs or chickens, to not think about the suffering they go through. Our job is to make them think. We're not out to be popular." Still, there's something impressive about the scope of the targets, that is, pretty much everyone. I tell Newkirk I was amused by the game "J-Lo Fur Bully on the Block" in which players had to help animals escape from Jennifer Lopez before they're turned into a coat or a pair of shoes. "Oh yes. We have lots of things like that. Have you seen Cooking Mama? That's bigger than all the others combined."
It's a parody of a Nintendo game, which shows a mother beheading animals to give to her children for dinner. Do mothers not get a bit cross about that?
"Of course they do. So they should stop being murderers." And then there are rather gleeful ad campaigns, including one that claimed a link between autism and milk. But it's based on such a tiny number of studies, I suggest.
"Yes, there are only a small number of studies," she says. "But that doesn't discount it. I'm not looking for a consensus. I'm looking for thought-provoking."
Isn't that just bad science? "It's not bad science. There's a link. Read the studies. Decide yourself. But every day people are told to drink milk, how it builds strong bones and so on. We don't have millions and millions of pounds to brainwash people, so we have our gimmicky thing. Milk has been linked to autism." And then there's her claim that "it's incontrovertible that not eating meat or dairy prolongs your life." Is it incontrovertible, I ask? You can be a pretty unhealthy vegetarian.
"There's study after study!"
But are they comparing like with like? Is a heart-healthy, low-fat diet which also contains a small component of meat really going to kill you first? "Fine. Then do it for ethical reasons. Do it for environmental reasons. We're not a health organisation."
There's something maddening about arguing with Newkirk, but then she's a provocateur. It's what she does. And it's what Peta, built in her image, does. It's why battery chickens are depicted in concentration camps and why last year she launched a legal case that named five orcas as plaintiffs and sued SeaWorld for enslavement. It failed, but she's surprisingly upbeat about it. The judge, she says, didn't simply throw it out, as he could have done. He was very "respectful" and heard the lawyers out. "It failed, but all the slavery cases fail when they're first brought."
Newkirk's argument is that if you're against slavery, it doesn't matter who is being enslaved. She is completely confident that one day we will look back on this as the dark ages. And, when she gets in full swing, describing the way that chickens are crushed en route to slaughter, their wings broken, the pain and inhumane conditions that they suffer, I find it hard to deny that she has a point.
What have you made of the horsemeat scandal, I ask her, and her eyes light up.
"There's this sentimental view that we don't want to eat horse. I've been in horse slaughterhouses, chicken, cow slaughterhouses, a dog slaughterhouse in Taiwan and not one of them wants to go down the ramp. They all kick. They all struggle. They're all petrified. It's purely sentimental." Would it have been better if we'd found puppies in our burgers?
"Oh God, yes. That would have been wonderful!"
Newkirk was born in England but spent the early years of her life being shifted from one school to another: Hertfordshire, the Orkney Islands, France and, finally, India and a convent school in the Himalayas - all cold showers and abusive nuns.
In the holidays, however, she'd help her mother in her work with the lepers, or the unmarried mothers, or the orphans. "She would literally have given you the shirt off her back. And she's always said it's not who suffers, it's how they suffer. And I've carried that with me my whole life."
She has. Newkirk's most oft-quoted statement and the encapsulation of her world view is that "a rat is a pig is a dog is a boy".
Her initial plan, however, after moving with the family to the United States, was to become a stockbroker. Then she had the encounter that would change her life.
