The quote provided by the OP shows where the author (Schilling) makes his own comments / conclusions / assumptions, and the inverted commas show the actual words of Luther. There is some alarming disparity between the two in the short clip provided. If the entire book was read, those apparent disparities might melt away, but given what little has been supplied, the following comments may be worth making.
First, "He made a promise of salvation to Christian soldiers" followed quickly with "This battle is a good and Christian work that can secure the soldiers’ salvation".
Those are not Luther's words. Until someone finds actual writings of Luther that plainly state such words, they cannot be taken as a fair summary of his meaning. On the contrary, Luther's writings abound with his stress on a person not being a Christian (even if he or she thinks otherwise) unless they have come to saving faith in Christ alone. But when Luther speaks of soldiers who are, indeed, Christians and find themselves in that vocation, he encourages them to do their duty as soldiers. That is not a promise of salvation, however. If they are Christians, their salvation is assured by virtue of what Christ secured for them, not because of engaging in warfare!
When Luther spoke about soldiers who were Christians, their salvation was already assured by soldiers with such faith, and those soldiers would have confessed and repented before baptism. There is not even a slight similarity here with Islamic promises of a virgin-filled paradise for soldiers, irrespective of anything other than them killing enemies of Islam in war.
Second, in Luther's time, he wrote according to the grievous political situations that led to war, and the religious situations that divided Christians on the matter of warfare. Here is a snapshot of a growing dilemma for Luther:
"In 1525 the Anabaptists in Zurich were subjected to the death
penalty. Luther was not yet ready for such savage expedients. But he
too was appalled by what to him appeared to be a reversion to the
monastic attempt to win salvation by a higher righteousness. The
leaving of families for missionary expeditions was in his eyes a sheer
desertion of domestic responsibilities, and the repudiation of the
sword prompted him to new vindications of the divine calling alike of
the magistrate and of the soldier." Here I Stand, p.268 Roland
Bainton, Lion, 1978 [bold emphasis mine]
Now, in light of the foregoing, view the following points Luther held to, regarding the state, politics, and soldiers who were Christians:
"When Luther came to construct a theory of government, he relied, as
in theology, on Paul and Augustine... Luther was perfectly clear that
coercion can never be eliminated because society can never be
The world and the masses are and always will be unchristian, although
they are baptized and nominally Christian. Hence a man who would
venture to govern an entire community or the world with the gospel
would be like a shepherd who should place in one fold wolves, lions,
eagles, ,and sheep. The sheep would keep the peace, but they would not
last long. The world cannot be ruled with a rosary.
The sword to which Luther referred meant for him the exercise of
restraint in preserving the peace both within and without the state.
The police power in his day was not differentiated from war, and the soldier had a dual function. In the use of the sword the ruler and
his men act as the instruments of God." (Ibid. p.238) [bold
Finally, your actual question is, "Where is the source of the quote in Luther's writing or any other place?" Martin Luther's words will be given source references in Schilling's book. Look for a small number following any quotes directly attributed to Luther, and find the corresponding number either at the bottom of the page, or the end of each chapter, or in "Notes" for each chapter at the end of Schilling's books. The number 87 is given for a few of Luther's own words; get Vol. 30/2 of the book reference stated, and go to pg. 180.