I am 56 and read (as a teen in the 70's) a short story
"Perchance to Dream", a short story by Sally A. Sellers, previously identified as the answer to the question Identify this story about a quickly healing woman; first published in Isaac Asimov's Science Fiction Magazine, Spring 1977, available at the Internet Archive. Does any of these covers look familiar?
about a woman who has no idea why she is immortal, but is now desperate and lonely and wants to die.
She closed her eyes. Her mother had never known. Had died before she realized what she had brought into the world. Before even Jeanette had an inkling of what she was.
A monster. A freak. This body was wrong, horribly wrong. It should not be.
She had run away from this town, left it so that her friends would never know. But still it pulled at her, drawing her back every generation, pushing itself into her thoughts until she could stand it no longer. Then she would come back to stare at the old places that had been her home and the old people who had been her friends. And they didn't recognize her, never suspected, never knew why she seemed so strangely familiar.
Once she had even believed she could live here again. The memory ached within her and she quickened her pace. She could not think of him, could not allow the sound of his name in her mind. Where was he now? Had he ever understood? She had run away that time, too.
She'd had to. He was so good, so generous, but she was grotesque, a vile caprice of nature. She loathed the body.
It was evil. It must be destroyed.
Here, in the city where it was created: Where she was born, she would die.
She tries to die, but she keeps waking up repaired and alive. Finally she gets killed (I do not remember how--I think she is stabbed interfering with a crime),
She does get stabbed interfering with a crime, but that happens at the beginning of the story:
"Okay, just do what we say and nobody gets hurt," snarled Danny. He pointed his knife at the boy's face. "You got a wallet, kid?"
The boy stared in mute terror at the knife. The girl made small whimpering sounds in her throat, and Norb tightened his hold on her collar.
"Come on, come on! Your wallet!"
From somewhere in the shadows, a woman's voice rang out. "Leave them alone!"
She recovers easily from that one:
The heart throbbed again, and another pinprick of light jumped behind the woman's eyelids. The tissues in the neck tightened further as new cells developed, amassed, and forced the blade a fraction of an inch outward. The wounds in the back, shallow and clean, had already closed. The lungs expanded once with a great intake of air. The knife jerked again, tilted precariously, and finally fell to the ground under its own weight. Immediately new tissue raced to fill the open area.
and her organs are harvested.
Crosby twisted away from her and moved to the window. No, he thought, we haven't much time. In a few minutes, she would get up off that table herself and walk into this room—and then it would be too late. She wanted to die. She had been trying to die for years—how many? Fifty? A hundred? If they took her organs, she would die. Not even that marvelous body could sustain the loss of the major organs. All he had to do was say yes. But how could he? He hadn't even seen her face yet. He could touch her again, talk to her, hold her. After thirty years!
[. . . .]
Grafton Medical Center was highly efficient. Within minutes, a surgeon was summoned and preparations had begun. The first organs removed were the kidneys. Then the heart. Later, the liver, pancreas, spleen, eyeballs, and thyroid gland were lifted delicately and transferred to special containers just above freezing temperature. Finally, a quantity of bone marrow was removed for use as scaffolding for future production of peripheral blood cellular components.
What had been Jeanette Crosby was wheeled down to the morgue.
In a bizarre ending twist, the story uses the point of view of her body, and we experience the struggle to rebuild all the organs, until right as the heart sets up to beat, the crematory oven kicks on.
Two men lifted the casket and bore it outside in the rain toward the oven.
Cells divided, differentiated, and divided again. The reconstruction was almost complete. It had been a long time, almost twenty-four hours. The body had never been challenged to capacity before. Removal of the major organs had caused much difficulty, but regeneration had begun almost at once, and the new tissues were now starting the first stirrings of renewed activity.
The casket slid onto the asbestos bricks with a small scraping noise. The door clanged shut, and there was a dull ring as the bolt was drawn.
There was a flicker of light behind the eyelids, and the new retinas registered it and transmitted it to the brain. The heart pulsed once, and then again. A shuddering breath.
Outside the oven, a hand reached for the switches and set the master timer. The main burner was turned on. Oil under pressure flared and exploded into the chamber.
There was a shadow of awareness for a long moment, and then it was gone.
The ending line indicates that even the ashes were large and hard to crush.
After thirty minutes, the oven temperature was nine hundred degrees Fahrenheit. The thing on the table was a third of its original size. The secondary burners flamed on. In another half hour, the temperature had reached two thousand degrees, and it would stay there for another ninety minutes.
The ashes, larger than usual, had to be mashed to a chalky, brittle dust.
In the epilogue, a doctor visits the girl who received her heart and finds the surgical wound already healed...
"But it isn't possible!" cried Dr. Kornbluth.
The girl spoke up in a high voice. "Is my new heart okay?"
"It's fine, honey," said Dr. Kornbluth. Then she lowered her voice. "This is physiologically impossible! The incision has completely healed, without scar tissue. And in thirty-two hours, doctor? In thirty-two hours?"