I am currently at a loss to explain how the suspect managed to delete the classified files. He claimed he did it by no article mistake, but I was under the impression that the system required a complex password and two-factor authentication.

at a loss: This is a fixed idiom meaning "uncertain" or "unable to know what to do or say," and it always requires the indefinite article "a".

by mistake: In English, many phrases starting with "by" that describe how something happens (e.g., by chance, by accident, by mistake) do not take an article.

under the impression: This fixed expression means "believing something (often wrongly)" and always uses the definite article "the".