[verb ない-form 〜ない → 〜] + ずじまいだ [verb ない-form 〜ない → 〜] + ずじまいになる [verb ない-form 〜ない → 〜] + ずじまいで、[sentence] する → せずじまい
ずじまい means "never got around to it": you meant to do something, the chance passed, and it never happened. Take the ない-form and replace ない with ずじまい – 言わない gives いわ言わずじまい, 見ない gives み見ずじまい. It almost always looks back on something already over, ending in だった or になった, and it carries regret.
する becomes せずじまい, and くる来る becomes こ来ずじまい, read こずじまい.With a potential verb, ずじまい says the chance never came, rather than that you put it off.
In the middle of a sentence, ずじまいで is "without ever having done it", and the rest says what happened next.
ずじまい is written and a little formal, though you hear it in conversation too. The plainer way to say it is けっきょく結局から~なかった or 〜~ないままおわった終わった.