Well, I wasn't wrong. But if you want to argue, here we go.
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Ashe could have selected nearly anyone for the king, but she chose Tryndamere.
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Even if Ashe doesn't care about Tryndamere, she will not leave Tryndamere because it will throw the Frelijord into a state of disarray for the rulers to break up and she cares very much about the Frelijord.
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Hardly any Frelijordians like Noxians because of the "Noxian Barbarian Pacification Campaign" an excuse for Noxus to send soldiers to the Frelijord and conquer it piece by piece, a plan which Katarina actively supports. Because of the discord it brings to her people, Ashe would most likely hate the "Pacification Campaign" the most.
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Their characters would clash horribly. While Ashe is a rather calm person that wants the best for her people, Katarina is literally "The Sinister Blade" that wants Noxus to conquer the world, Frelijord included.
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Katarina is an assassin, Ashe is a high-priority target, and not just in-game. Because Ashe is the queen of a rival city-state, Noxus would likely want her dead and I know that Katarina would likely oblige.
I could go on forever, but I know that I'm just going to receive an onslaught of downvotes for respecting the lore and, more importantly, nobody's going to listen. You're going to believe that Katarina and Ashe will make googly eyes over a sundae and I think that Ashe's calender will be filled with too much Frelijordian business to date Noxian assassins and nothing short of death will deviate our opinions from that.