In the Shabbos morning Tefillah, the Tehillim 136 is said, which includes the above verses. First, the tehillim mentions the destruction of the mighty kings, and then afterwards, mentions in particular Sichon and Og, as exemplifiers of the strong and fearful kings. Why are these two kings singled out from among all the kings that were defeated?
The explanation is contained in the concept that not all people opposing the Jews were fitting that the Jews would occupy and inherit their land. As can be seen in the second chapter of Devarim, many lands (Ammon and Moav) were not to become occupied lands of the Jews despite the military victory. However, these lands did become “purified” by the hand of Sichon (Gittin 38a).
As explained in Chullin 60b, even though HaShem stated that the Jews were not to conquer the lands of Ammon and Moav, but when Sichon came and took land from Moav, and then the Jews by defeating Sichon, came to possess lands that had belonged to Moav. Similarly, Og took land from Ammon, which after defeating Og, the Jews assumed possession of this land. Thus, Sichon and Og enabled the lands of Ammon and Moav to become into the possession of the Jews.
And now we can understand the order of the Tehillim.
Refers to the kings like Ammon and Moav, that the Jews defeated but did not gain any material advantage from this victory
למכה מלכים גדולים
After defeating these two kings, who had previously took lands from Ammon & Moav
למכה סיחון... ועוג
And then HaShem (in addition to the lands of Sichon & Og) also provided (some of) the lands of Ammon & Moav
ונתן ארצם לנחלה
