てしまう
te-shimau
てしまう is not regret. It is a lid closing.
The picture You put it in a box and shut the lid. It cannot be reopened — which is both 'all done' and 'well, that's that'. ておく leaves it open for later; てしまう shuts it where you cannot reopen it.
Where the picture shows up in てしまう
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done to the end and shut
レポートは書いてしまいました。
repōto wa kaite shimaimashita
I've finished the report.
The report, printed to the last page, goes into the box and the lid comes down.
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shut, and no way back
財布を落としてしまった。
saifu o otoshite shimatta
I lost my wallet.
The wallet drops into the box, the lid shuts, and your hand cannot reach it.
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the spoken short form
電車に傘を忘れちゃった。
densha ni kasa o wasurechatta
I left my umbrella on the train.
The train doors close with the umbrella still inside, and the train pulls away.
In speech ~てしまう becomes ~ちゃう and ~でしまう becomes ~じゃう: 飲んじゃった.
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emptied out and shut
ケーキを全部食べてしまった。
kēki o zenbu tabete shimatta
I ate the whole cake.
The cake plate is bare and the empty box snaps shut.
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and so it turned out
寝坊して、遅刻してしまいました。
nebō shite, chikoku shite shimaimashita
I overslept and ended up late.
The lid comes down over a clock that slept through its alarm, leaving only 'late' behind.
When apologising, 遅刻してしまいました sounds far more natural than 遅刻しました.
All of it in one picture
てしまう is neither regret nor completion; it is one picture of a lid coming down on something you cannot reopen. Finish the report and close the lid and it feels like an accomplishment; let the wallet fall in first and the very same lid is a disaster. The train doors shut with the umbrella inside; the plate empties and the box snaps closed; the lid drops over a slept-through alarm and only 'late' is left. Good or bad, the moment てしまう attaches, the thing is already shut.