。
She somehow managed to escape being found.
Literal
She [topic-は] somehow [without being-found-out-ずに] got-by.
見つけ出す is a compound verb ('find + out' — search and locate); 見つけ出される is its passive ('get found out'). The frame [verb-negative]＋ずに済む means 'manage to get by without (having to) X' — used when an unwanted action or outcome was avoided. Stitched together: 'managed without getting found out.' どうにか reinforces that the escape was narrow or required effort. The literary ~ずに is the negative gerund, equivalent to ~ないで but stiffer and more written-leaning. Together the sentence has a slightly noir, stakes-y feel.