-kiru is an auxiliary verb that shows you did something completely.
Formed from Verb -masu + kiru Note that 'kiru' conjugates like a normal verb (e.g. past tense, negative etc are possible) Vm = Verb -masu base
is this somehow related to shimau which also means to complete?
I believe kiru is more informal and prose in nature than shimau.
The difference between shimau and kiru is not a matter of informal and formal, I don't use the words in that way as a native.Their nuance are different and they are used in a different way. Kiru would be one intend to finish doing something "to the end." Shiamu would be one finish something, "sometimes" with the nuance it was against one's will. ハリーはなんとかその川を泳ぎきった。 ex4654 Harry managed to swim across the river( with his intension.) ハリーはなんとかその川を泳いで渡ってしまった。 Harry somehow managed to swim across the river. ダイエット中のくせに彼女はケーキを食べきりました。 ex #1075 Even though (despite the fact) she was on a diet, she ate all of the cake. (ate it to the end) . (She ate all knowing what she was doing.) ダイエット中のくせに彼女はケーキを食べてしまった。 (I don't know she ate cake all or not, anyway) although she is on diet, she ate a cake (as she could not stop herself.) 夏休みが前に、宿題をやりきろう。 Let's finish all the homework before summer vacation starts. 夏休みが前に、宿題をできるだけやってしまおう。 Let's finish homework as much as we can before summer vacation.
夏休み前に、instead of 夏休みが前に、
きるalso shows that something has reached its limit. ex #6034 and ex #6035
Miki? 夏休みの前にと言えば いいじゃないか？
4655 seems to be a slightly different case, 'iikiru' 'to declare ouright', rather than any meaning of doing something to the end.