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How shall we say "不甘心“ in English? For example, "经历多次失败后，仍然不甘心放弃过去的种种努力和自己的理想”.
Note that "is not willing to" means "不愿意“， still not exactly "不甘心“. Consider an example, "经历多次失败后，仍然不甘心放弃自己的理想”， “不愿意帮助他”。 In the first example, “不愿意“ doesn't have the exact meaning of "不甘心“, while in the second example, “不愿意“ cannot be replaced with "不甘心“.
How about "not resigned to" or "not happy to" for 不甘心?
In the right context, it can easily be translated as or ‘not willing to’. In other contexts, different translations may be necessary.
In your example, ‘not willing to’ does seem stronger than 不甘心 implies, though ‘unwilling’ might work. Better than ‘unwilling’, though, I would suggest either ‘hesitant’, ‘loath’ or ‘reluctant’:
经历多次失败后，仍然不甘心放弃过去的种种努力和自己的理想。
Despite having failed several times, he was still hesitant/loath/reluctant to change his views.
不甘心=(litterally)to not make heart sweet=not willing to feel content
=（extensively）not willing to have a feeling of being a loser / not willing to accept the current bitter situation
I don't deserve that.
不甘心 means you work hard and don't get reasonable feedback.
