This Wednesday, The Culture Show is running a half-hour special on the pop phenomenon that is Lady Gaga, hosted by me. This is considered as something of a scoop. Despite Gaga's willingness to promote her new album, Artpop, she is hard to pin down: she is wanted by so many media outlets, from Graham Norton to The X Factor to the Daily Mail. Anyway, BBC2 is very pleased. And so am I, I think. I don't really know. It was an odd experience.

Before we met, I did a lot of Gaga research. I watched her talk show chats, revisited videos and songs, her performances on awards shows. I read broadsheet pieces, tabloid splashes, studied her Twitter feed. I did this because it's part of the job, but also because our interview kept being cancelled, sometimes as late as 8 o'clock the evening before. There wasn't much else to do but go back to the research. Here's some, so you don't have to bother.

Lady Gaga's first proper single, Just Dance, came out in August 2008. Since then she has sold more than 23m albums and 64m singles; her Born This Way Ball tour grossed $382.3m; she has more than 40 million followers on Twitter. For Artpop, her third album (other LPs: The Fame and Born This Way, plus The Fame Monster EP in between), as well as collaborating with people like R Kelly and rapper TI, she has been working with three very different contemporary artists. There's po-faced performance artist Marina Abramovic, revered for her lengthy, emotional/non-emotional presentations and for somehow getting Jay Z to perform Picasso Baby for six hours straight this summer. Then there's equally serious theatre director Robert Wilson, who was in charge of Gaga's quick-change performance of Applause for the VMAs and who makes video depictions of celebrities called Voom Portraits. And finally – the weirdest one for me – there's Jeff Koons, known for his vast, kitschy sculptures and his unparalleled ability to make money. (I once interviewed Italian porn star La Cicciolina, who was married to Koons for a few years - he made sculptures of them having sex. She told me he stayed in bed all day watching videos, and his bath towels were tatty: "I no like big hole towels," said La Cicciolina, with some force.)

Anyway, after much to-ing and fro-ing, The Culture Show team became convinced that the interview wasn't going to happen, so we started making a different programme, called Waiting for Gaga. We hung out with her fans, the Little Monsters, who were gathered in a devoted mob around the entrance to the Langham hotel. They were young (14-24ish), articulate and had certain rules (the hardcore got very upset when some other fans entered the hotel: "She needs her space!" they cried). I was told that "paws up" – the Little Monster fan sign, two hands grabbing at the air – was seen as a bit naff these days. And that they preferred hanging around the Langham, almost more than they did going to her gig.

Observing her (from close up; but, really, from afar), I could see why. When Gaga leaves or arrives at her hotel, she doesn't just rush, head down, sunglasses on, from people carrier to lobby, blindly scribbling autographs on anything thrust under her nose. Instead, she stages a small show. She is in character, she has a look that she inhabits, like an actress. This look might be deathly, or sexy, or – as recently, in Berlin – she might be sporting a triangular feathered box on her head. Sometimes she poses for "selfies". A selfie is every Little Monster's ultimate goal.

We were quite happy with our new fan-angled programme, when we were given notice that we definitely had a Gaga interview. Friday morning, just before she flew back to the US. I asked to hear the album; I was told my listening slot would be two hours before our chat.

So after hearing Artpop at full blast in a studio (there are at least six fantastic songs; the first half is better than the second, and the ballad, Dope, is reminiscent of Meatloaf), I hopped in a cab to the hotel. In the allotted interview space, a meeting room swathed in white fabric to cover the swirly carpet, the atmosphere was cheerful, but tense. Would this interview finally happen?

Gaga has a preferred lighting team – old-school British sparks, who talked about football and their grandchildren – and she likes to have things checked before she arrives: the chair height, the camera shot, the ambient temperature. Gradually, the room filled with more and more people. Waiting. How strange to be so famous that you're like a starting pistol: wherever you go, nothing happens until you walk in and set off the race. I talked to a man from her record company. He told me that, at the moment, Gaga was all about the positive. That Born This Way was about accepting yourself as you are, and Artpop is about helping the fans put creativity and positive energy out into the world. I tried to square this with the lyrics to Donatella, on Artpop: "I'm blonde, I'm skinny, I'm rich and I'm a little bit of a bitch."

Lady Gaga with tearstains. Photograph: Beretta/Sims/Rex

And then, she was here. Like most celebrities, smaller and more beautiful than you imagine, Gaga materialised as silently as a ghost, and dressed in a similar style. An off-white, raw silk, floor-length gown. No shoes. Arms streaked chalky white; face smooth and porcelain pale; eyebrows whitened so they merged with her skin. Burgundy lips and dark waved hair, eyes all smudge and shadows. Miss Havisham meets Ava Gardner.

She sat. Checked her image in the monitor. Asked someone to unzip her dress at the back so she could breathe, and we began.

I asked a benign opening question about how she felt to be back, having been forced to take time off (she had an operation on her hip). She answered honestly, interestingly – "You feel like an infant… I think it was good for me, because the stage had become a place I had started to rely on" – but in a flat, deathly monotone. She had been working with Robert Wilson for the past 48 hours, she said, recreating certain art images (The Death of Marat, The Head of Saint John the Baptist). She hadn't slept. I pressed on. She told me she was having great sex, that her earlier sexual experiences "were quite perverted and scary, terrifying". She called pop music "the cum shot" and said she didn't even know if her music was any good. She said, "I don't find myself that sexy actually." I wondered how much art could truly be embedded in her music. She said that all her learning about art ensured that it would come through. At least, I think that's what she said. That answer took a long time.

We spoke for 40 minutes, and around halfway, she stopped completely, holding a pose for 30 seconds, saying absolutely nothing at all. The room, still full of people, froze with her.

In the end, it was an encounter, not an interview. She controlled it all, and her mood – tired, overly intellectual; full of art, rather than pop – was what dominated. Despite this, I liked her. She was clever and thoughtful. She just wasn't at all like any of the interviews that I'd researched: she didn't gush, she was serious and still. "I might sound total shit," she said. "Your ratings will go down."

Miranda Sawyer's selfie with Lady Gaga. Photograph: © Miranda Sawyer

When we finished, the director asked if we could shake hands, to round things off. "Let's do a Marina hug," said Gaga, and we stood, me towering over her, in an odd embrace. I could feel her breath shuddering through her body. She shook everyone's hand ("That's new," remarked the lighting producer, sotto voce), gave me a selfie and then said to her people, "Do we have a thong?" No thong. So she wriggled out of her knickers so as not to ruin the line of her dress, stood under the hot, bright TV lights and asked us all, "Can you see my thing?" No. You're good, Gaga. No "thing". Nothing. She smiled.

And then, she left. Dematerialised from the room to pose outside the Langham, walking slowly forward in her bare feet, as though in a trance, as though her limbs were heavy. The fans shouted; the paparazzi snapped and snapped. If you looked closely at the pictures in the papers the next day, you saw that her cheeks were streaked with tears.

The Culture Show: Lady Gaga is on Wednesday at 10pm, BBC2