The Median of Death
The first page of the chapter is dedicated to explaining what a "lethal dose" is - the amount of a substance that the body can intake or can exist in the body before it becomes lethal to the target. Once that limit is exceed, you will die as a result. It also explains that there is a "Median" for the "lethal dose", which is the amount required on average to kill 50% of a tested group of people, followed by a number of examples, which includes the puffer fish's poison, nicotine and even Vitamin C.
At this point in time, this seems to be the second ability in Bleach that is directly related to a scientific concept (the other being Rukia's Sode no Shirayuki), so it is interesting to see Kubo bring science into the equation regarding powers in Bleach once more.
Dealer of Death, Drinker of the Lethal Dose
With the explanation given, we cut back to Askin who explains the detail of just what "The Death Dealing" (written in the RAW as "Lethal Dose") is and how it works. Based on what Askin is saying, the power automatically calculates the 100% "lethal dose" whenever a substance is "absorbed" (going off Mangastream), and he can freely manipulate that amount as he sees fit.
Initially, Nimaiya does not quite understand what Askin is talking about, but Askin states that it doesn't matter because Nimaiya is about to die anyway, and we see Nimaiya fall to his knees as Askin licks his own blood. He then proceeds to narrate how he lowered the "lethal dose level" for the amount of blood Nimaiya's body can safely hold before it becomes fatal. He goes on to cryptically state that the same goes for him as he proceeds to drink a mouthful of blood from his hands.
Looking at how this scene played out - Nimaiya collapsing further the more blood Askin drank - I do believe we are seeing the mechanism by which "The Death Dealing" works. In order for Askin to manipulate the "lethal dose level" of a specific substance in an opponent, he first needs to ingest that substance and introduce it into his own system. In this case, he ingests the blood spilling from the wounds Nimaiya inflicted on him to target the "lethal dose levels" of the blood in Nimaiya's body.
I do not know how it would work with other substances, but that seems to be how Askin tried to take out Nimaiya in this chapter.
Die of Too Much, Die of Too Little
As Askin finishes explaining how he thinks Nimaiya underestimated Askin because of the personality he gives off, Nimaiya thanks Askin for explaining his power, cuts his own neck and proceeds to try to attack Askin again. Nimaiya explains what he understood from Askin's description of "The Death Dealing", and decided to cut his own neck to reduce the amount of blood in his body down to levels that would not be lethal.
However, Askin points out it is not that simple. After a short banter about the approximate ratio of blood in the body per kg for someone of Nimaiya's weight - and the point where someone dies if they lose too much blood - Askin comments that he reduced the "lethal dose" of the amount of blood Nimaiya can safely lose to 1.6 Litres.
This really shows how dangerous Askin's power actually is. Even if you were to try to bleed yourself out to reduce the amount of blood in your body down to a point below the "lethal dose" threshold set out by Askin's "The Death Dealing", he also manipulates the opposite end of the spectrum - how much blood you can lose before you die.
In any other situation, Askin would have legitimately had Nimaiya on the ropes.
Hot Spring Demon and the God of Swords
Unfortunately for Askin, Nimaiya calls for Tenjiro to provide aid. As revealed, while the water Kirinji used initially was the White Bone Hell water, Kirinji is using the Blood Pond Hell water to replenish and replace the blood Nimaiya lost.
A possible reason why this would nullify Askin's ability is because the Blood Pond Hell water is now mixed in with Nimaiya's own blood. The target substance of Askin's ability did not possess that kind of composition, so "The Death Dealing" would no longer have a hold on Nimaiya's body as a result.
Looking at how the battle in the Royal Realm is turning out so far, it is nice to see how the Royal Guard members are combining their unique abilities to aid each other and to fight against Yhwach and Co. You really get the idea that the Royal Guard are a group that works together to combat threats. This is in stark contrast to the Gotei 13, who are generally independent of one another.
In this situation, Nimaiya would have been finished if he did not have Kirinji backing him up.
Death Comes to the Death Dealer...?
Now we reach the point that has gotten people up in arms - Nimaiya seemingly kills Askin. After Nimaiya strikes, Askin thinks back to when he thought that Nimaiya had underestimated him, and calmly realizes that he was the one who underestimated Nimaiya.
It is rare for a villainous character to calmly accept their defeat in such a manner in any media, and it can be a powerful moment when it does occur. In the case of the Stern Ritter, Gremmy and Askin are two notable aversions of villains completely breaking down as they die, as the two of them calmly realized why they were defeated and accepted their deaths. They neither begged for mercy, nor went down screaming, not comprehending that they have lost; they simply accepted death as it came.
In this case, however, the question is whether or not Askin is dead this time. After all, Nimaiya did not kill him successfully in the last chapter, even though it looked like he cut his throat open. It is likely that Askin's power can keep him alive through the same method it uses to kill - it increases the "lethal dose level" of Askin's body, meaning he can lose more blood than the average person before it becomes lethal, or something along those lines. As he stated last chapter, Yhwach took him in because "he just wouldn't die", and that what Nimaiya did to him was nowhere near enough to kill him. As a result, it is entirely possible that the blow Nimaiya hit him with this time is still enough.
Either way, Nimaiya is confident that he can now take on Yhwach without interruption, who seems to be preparing for battle himself.
As to whether or not Askin is dead, I suppose we will have to wait for the upcoming chapter to see what the end result is. Given that one of the Quincy among the Elite Guard is "The Miracle", we might get a come back of the entire Elite Guard.