〜かいもなく
despite (all the effort), in vain
Verb た-form / Noun + の + かいもなく
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かいもなく says an effort failed to pay off: 努力のかいもなく、試験に落ちた, "despite all my efforts, I failed the exam".
It's the negative of かいがある (N2), "it's worth it, it paid off". かい (甲斐) means "the worth or reward of an effort". The first half is an effort (studying, treatment, cheering, persuading), and the second half is a disappointing result.
It's a sad, sympathetic expression, often used in reports of illness and defeat: 懸命な治療のかいもなく、彼は亡くなった, "despite every effort to treat him, he passed away".
Compare おかげで, "thanks to", which is the grateful opposite.
Examples
努力のかいもなく、試験に落ちた。
Despite all my efforts, I failed the exam.
懸命な治療のかいもなく、彼は亡くなった。
Despite every effort to treat him, he passed away.
必死に応援したかいもなく、チームは負けた。
Despite our desperate cheering, the team lost.
毎日練習したかいもなく、一回戦で敗退した。
All that daily practice was for nothing. We went out in the first round.
何度も説得したかいもなく、彼は会社を辞めた。
Despite all my attempts to persuade him, he quit the company.