If you read Dan.11:36-45, the main topic of those scriptures is the Anitchrist (the son of perdition, the man of sin, the beast),...and not the king of the north or the king of the south.
36 "The king will do as he pleases. He will exalt and magnify himself above every god and will say unheard-of things against the God of gods. He will be successful until the time of wrath is completed, for what has been determined must take place.
-Notice the "king","he will exalt himself above every god" which is synonymous with all other scripture about the Antichrist.
37 He will show no regard for the gods of his fathers or for the one desired by women, nor will he regard any god, but will exalt himself above them all.
-Ok this is important where it says "he will show no regard for the gods of his fathers" which means that he will be Jewish. Once again at the end of that verse it reiterates that he "will exalt himself above them all". Whose them? The Jewish gods and any god for that matter, including Jesus.
38 Instead of them, he will honor a god of fortresses; a god unknown to his fathers he will honor with gold and silver, with precious stones and costly gifts. 39 He will attack the mightiest fortresses with the help of a foreign god and will greatly honor those who acknowledge him. He will make them rulers over many people and will distribute the land at a price.
Ok verse 38 and 39 explain that he will be a king of war and money.
Keeping all of these verses in mind and in context,...now lets read verse 40 and further:
40 "At the time of the end the king of the South will engage him(the Antichrist) in battle, and the king of the North will storm out against him(the Antichrist) with chariots and cavalry and a great fleet of ships. He(the Antichrist) will invade many countries and sweep through them like a flood.
-Regarding these verses it's important to understand who "he" and "him" is. "He" that is mentioned in verse 40 is still talking about the Antichrist of the previous verses. "The king of the South will engage" the Antichrist in battle,...and "the king of the north will storm out against him" (the Antichrist) too. "He (the Antichrist) will invade many countries and sweep through them like a flood."
-So in other words,...a king with army will be attacking the Antichrist in Israel from the south and a king with an army will be attacking the Antichrist in Israel from the north.
The next verses until the end of the chapter go on to explain more about what the Antichrist will do.
41 He will also invade the Beautiful Land. Many countries will fall, but Edom, Moab and the leaders of Ammon will be delivered from his hand. 42 He will extend his power over many countries; Egypt will not escape. 43 He will gain control of the treasures of gold and silver and all the riches of Egypt, with the Libyans and Nubians in submission. 44 But reports from the east and the north will alarm him, and he will set out in a great rage to destroy and annihilate many. 45 He will pitch his royal tents between the seas at [ f ] the beautiful holy mountain. Yet he will come to his end, and no one will help him.
-Once again,..."he" (the Antichrist) will invade many lands,...but for some reason Edom, Moab, and Ammon will be delivered. Edom, Moab, and Ammon is modern day Jordan,...specifically the Petra area in Jordan. Many think this will be the place God prepares for Israel when they escape the Antichrist and Jerusalem. (That's beside the point though.)
-Next,...Egypt will not escape the armies of the Antichrist. So this is only speculation but Egypt could be a part of the "king of the south" forces.
So who is the king of the south and who is the king of the north? It doesn't matter! It's a moot point. It's all speculation. Yes,...proplog, "the king of the north" may be Russian forces and the "king of the south" may be US forces,...but does it really matter? Plus I don't think it's right to say that the "king of the south" is absolutely this for sure and the "king of the north" is absolutely this for sure,...without a doubt. I think if we do that then we are playing the WT's game.
What really matters is to understand these scriptures in their context and that the context is all about what the Antichrist will be doing. The king of the north and the king of the south will be coming against the Antichrist,...and not necessarily against each other. Also, they will be invading Israel where the Antichrist will set him self up.