contribute A to B(infinitive or gerund)

"to help" is better

https://www.ldoceonline.com/ko/dictionary/contribute
https://dictionary.cambridge.org/ko/%EC%82%AC%EC%A0%84/%EC%98%81%EC%96%B4/contribute
What do you think of the explanations of the dictionaries?

They're all good explanations but they don't specifically apply to the question you are asking because the construction of your sentence is a bit more complex.

Does "help" sound more natural to you/native speakers than "helping" does in the above sentence?

Either one is acceptable, both are understandable. I feel that the infinitive (to help) is more direct and denotes action.

1) Hoping to contribute her skills to help the British troops fighting in the Crimean War, Nightingale accepted a nursing position at an army base in Scutari, Turkey in 1854.

Sometimes it can be helpful to alter the sentence structure:

2) Hoping to contribute her skills, Nightingale accepted a nursing position at an army base in Scutari, Turkey in 1854 to help the British troops fighting in the Crimean War.

"Helping" would not work in example #2.