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.
Harry managed to swim across the river( with his intension.)
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.)
(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.)
Let's finish all the homework before summer vacation starts.
Let's finish homework as much as we can before summer vacation.
4655 seems to be a slightly different case, 'iikiru' 'to declare ouright', rather than any meaning of doing something to the end.
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