Classically, Judaism has not made the distinction between body and soul. That is not to say that we do not believe that there is a difference between the physically tangible body and the highly intangible life-force, but Jewish tradition does not posit that they have existence independent of each other. You may think that's nonsense, given the respect with which we treat physical remains after a death, but it is no different than the way we treat sacred books, ritual objects or even tallis bags when they are no longer functional. They are not considered to be complete anymore, in spite of their resemblance to what they once were.
However, just as we have words for our physical body (guf, and the various body parts), we have words for the intangible part of our existence. One of them is "nefesh," but it is not the one that is used here, and is therefore the subject of a different discussion. The other one is "n'shama." And when it is translated as "soul," that translation gives a misimpression of what the word means.
N'shama is unambiguously derived from the Hebrew term that means "breathe." And in case there is any doubt about its meaning as life-force (as opposed to mere respiration), you need only look to the very beginning of Torah. In the story of creation, God gathers dust to form the first earthling into which God blows "nishmat chayyim," the breath of life. It is this n'shama that is the animating force in our physical bodies and without which we are incomplete.
You can no sooner separate a body from its breath than you can separate a breath from its body. They are mutually dependent and unidentifiable as whole without each other. And never mind that science has found ways to see a breath or to measure the weight of a soul; those statistics are still dependent on the wholeness of the two. We have not been able to isolate n'shama in any sense, nor would we know what to do with it (any more than with a corpse) if we did.
So how can we speak of "nishmoteinu, our souls" in this prayer and others? The use of the word recognizes that what is mysterious to us is understood entirely by God. It is a matter of trust that recognize that the gift that was breathed into us at creation is breathed into us at every moment, and a matter of gratitude that what was given once is given every day, held by God in that space between every inspiration and every expiration – in the many senses of those words.
Were I to leave things here, you would likely walk away convinced that Judaism, or at least Jack Moline the Jew, would not affirm life after death. So I tantalize you with a teaser for another time and place – don't believe it for a minute.