Suppose we are to believe that the soul exists. If the body is extinguished upon
This is a terrific question, even though you must admit that you are assuming all sorts of things: above all that the soul is the functioning of the body, and that the body is extinguished with death. One important tradition in thinking about immortality of the soul refuses to accept the former of these. That is the Platonic tradition, which believes that it can conceive of and define the soul independently of this material functioning. On that tradition, the puzzle you are raising does not come up.
However, the tradition that begins with Aristotle does indeed understand the soul as your question assumes, and there are even strains within Platonic philosophy that do the same. Certainly Christianity, with its belief in the resurrection of the body, proceeds with the thought that immortality must be grounded in the body's continued existence. So let's acknowledge that your assumptions are not shared by all theorists on the subject and press on.
One question the defender of immortality might still press you on has to do with your claim that sharpness does not survive the destruction of a knife. Surely that's false. One knife goes, but many others survive. No more of that knife; plenty more sharpness still around. Get rid of all knives and razors and physically sharp things, and yes you will thereby get rid of sharpness. But it's not as easy as banging up a single knife. What I think you mean to assert, in other words, is that sharpness (and therefore soul) requires some physical instantiation. Let's grant that. It doesn't follow that sharpness (or soul) requires instantiation in a particular object. Why can't there be another body in which my soul persists after the extinction of the body that defines me now?
Again, the Christian tradition apparently believes just such a thing; hence remarks about the body of the resurrection. And even that contrary thinker Plato can be interpreted to agree with this tradition. As early as the Cappadocian philosophers, in the generation before Augustine, we have what purports to be Macrina's deathbed discussion of immortality (as either transcribed or composed by her brother, Gregory of Nyssa). Macrina refers to the reincarnation or metempsychosis found in most of the Platonic dialogues' discussions of immortality. According to Plato (usually) the soul can pass from one human body to another, or even to the body of another species. Macrina would not make either assertion, yet she thinks Plato is on the right track. For what reincarnation acknowledges, according to her, is the need for souls to persist in some body or other. Plato just didn't see that there could be a unique body for one's soul to go into after the nonexistence of the present one.
These suggestions are not the end of the story. The conversation needs to go on at great length after Macrina/Gregory, and contemporary Christian theologians, among others, have taken it much further. My only point was to suggest to you that even assuming a view of the soul like the one you hold, there is a way of conceiving of its existence after death.