I think this story could be "World Without End" by F. Gwynplaine MacIntyre, published in 2010, in two different Mammoth books: The Mammoth Book of Apocalyptic SF and The Mammoth Book of the End of the World
It's a story told in first person from the journal of a 16 year old girl who was injected with nanobots as a guinea pig for an experiment. She was a street prostitute originally. The sample excerpt from Google Books goes: "I was a street bint: selling my snatch to kerb-crawlers and pervs...".
She was injected by a university student unwillingly post-handjob:
"One of those clever lads from the Uni, he was, reading science. While
I was pulling him off in the alley, he kept nattering to me about
something he was working on. Nanotech, that was the word. And summat
called the Hayflick limit, which meant bugger-all to me at the time,
only I sussed it out later. Said he hadn't kept track of his own
experiment, and wasn't sure he could ever repeat the results. I wanted
him to stop jawing so I could just concentrate on getting him off and
then get on to my next punter, but he kept blagging about little tiny
robots and such. I was wearing a skimpy miniskirt and a halter. He saw
the tracks all in the veins in my arms and my legs and he knew I was
on the stuff. Then he jabbed a needle into my arm, injected something.
It felt like fire in my blood, ooh I wanted it I wanted it."
As for slang for her lady bits: "They did tests on my fadge, and that, several times my lady-bits melted the tips of their scientific thingies." Apparently her fadge becomes blisteringly hot post injection....
After she is rendered immortal, she witnesses the end of humanity to a Great Pandemic which eats flesh and bone leaving behind piles of goo, all the humans and most animals died, except for insects. This is when she walked right on out of the cell she was being kept in for study.
She sees at least two different alien races come to the earth afterwards, the "Bug-men", and shiny green aliens.
At some point, the healing factor of the nanotech gets to the point where she's becoming less than human.
"It's just so bloody not fair. I can't eat I can't drink I can't sleep
I can't breathe and now I can't even cry. I've lost everything that
ever made me human except I can't die."
She's been alive for at least 3000 years, and she's watching the sun get bigger and redder and hotter. She hope that when the Earth finally melts or drops into the sun, it'll kill her and she'll have peace at last, but at the same time she fears that she'll still be alive in the sun burning and dieing and healing infinitely. Eventually the seas dry up and her skin starts burning and rehealing and burning over and over.
The last lines:
"At last at last billions of years falling into the sun hurry up will
I die now or will it get worse now I'll know if
oh damn its not fair"
Cringe worthy fate worse than death.