Ivan falls ill after a tumble in his drawing room; Ivan grows up and starts leading a false life.
There are two good candidates for the moment when Ivan "falls under The Dark Power," each of which gives a different reading of "The Dark Power" which overwhelms Ivan. One possibility is his literal fall while doing the drapes in his drawing room, after which he becomes ill. From then on, Ivan's pleasant life stops being pleasant and his mood sours permanently. This reading takes Ivan's illness and death as the dark power.
But then again, as Ivan will realize in his last moments, it's only because Ivan's whole adult life was so false that death appears as such a terrible thing to him. So you could also say that death frees Ivan from the real dark power, which is his false life (See "Character Roles" for more on this). If you read the novella this way, then Ivan's fall happens sometime in the very condensed narrative of his life story, whenever he leaves his happy childhood behind and starts living as an adult in the false world.
We'll call the two readings the "Illness/Death" reading and the "False Life." They'll result in two different ways of dividing up the plot. (And in case you were wondering, since the Rebirth narrative is all about Ivan's rebirth, the first chapter is kind of an odd one out, since Ivan's not in it. It just serves to provide a preface by showing the darkness – falseness – of the world in which Ivan lived).
Ivan hopes for medicine to save him; Ivan lives a pleasant, proper, and very normal life.
On the "Illness/Death" reading, Ivan's recession stage is that early period of his illness when he hopes that medicines and doctors can cure him. It's not a recession, technically speaking, since it's not as if things ever get better. Things just haven't hit really bad yet, since Ivan hasn't realized that he's doomed.
On the "False Life" reading, Ivan's recession stage is his entire adult life. Although the fall has taken place, it's not experienced as such by Ivan; everything in his life from his point of view seems just peachy. No matter if his marriage is a failure or his life is meaningless, he's always got bridge and work to distract him. Only after he's well into his illness will he realize his life has been a bad one.
Ivan loses his hopes for a recovery and is forced to confront his own death, alone; Ivan falls ill and his old, pleasant life fades away.
In terms of the "Illness/Death" reading, Ivan's imprisonment begins as soon as he recognizes that his illness can't just be fought off with medicines. He realizes that it's "something, new, terrible and more important than anything before in his life" (4.17). Now everything in his formerly enjoyable life becomes colored by the prospect of death, and he's left to deal with it all alone.
On the "False Life" reading, the imprisonment stage begins when Ivan falls ill. This is when his mood takes a permanent turn for the worse, and when his life stops being pleasant and easy. The deepening of the darkness over Ivan's life is gradual, from the early days when Ivan still has confidence in medicine through to the period when death confronts him everywhere he turns.
Ivan's last, faint hope of recovery deserts him after the celebrated specialist's visit; Ivan recognizes that his whole life might have been a mistake, and can no longer desire to return to it, even though the present is horrible.
Technically, the nightmare stage is supposed to be when things look really hopeless. As hopelessness tends to be the norm in the last part of Ivan Ilych, it's a little hard to place this stage specifically in the novella.
In terms of the "Illness/Death" reading of the story, the morning the celebrated specialist visits (Chapter 8) is a good candidate, since by that point Ivan seems to regard his death as inevitable and experiences his life as a monotonous torment. He doesn't want to live, but he doesn't want to die either. The specialist's visit itself gives him a last, very brief moment of hope that he might recover. This hope vanishes once the specialist leaves, after which Ivan is totally hopeless.
A possible Nightmare Stage candidate for the "False Life" reading is the moment when Ivan first considers that his whole life may have been wrong. Before this moment Ivan wanted things to return to go back to the way they were. Now even Ivan's old life is revealed to be horrible. He has nowhere to turn to, except back to his distant childhood.
We should note that by this point two readings of the novella are starting to converge. For both, the darkest period of the nightmare stage (and the novella) is Ivan's three days of screaming.
Ivan falls through that narrow black bag to find miraculous light.
The rebirth is supposed to be a "miraculous redemption," and that might apply literally here. Just when things look like they couldn't get any worse and it seems impossible that the screaming, tormented Ivan can do anything to save himself, Ivan is suddenly enveloped by light, and is freed. Both dark powers – death, and Ivan's former life – are defeated in the same stroke. Ivan calmly rejects his old life in order to redeem himself in his last moments and finds that his fear of death has fallen away. Since they fall together, the two "dark powers" are revealed by the end to be closely connected. Ivan's life up to this point has been a kind of living death. His death was an enemy because his false life did not provide him the resources to deal with it. Liberation from one means liberation from the other. Ivan's abandonment of his false life means that "Death is finished" too.