Is it considered a violation of Islamic law for a brother to pay for his brother's laundry from the account of the club where he works?

If the brother paid the laundry fee from money he did not own, then this is not permissible for him because it is an encroachment upon another's property. You must pay the amount to the club from which he took the money. However, if the brother's money came from an illicit source: if all his money is unlawful, it is not permissible to benefit from it, and you must pay an amount equivalent to the fee and spend it on public welfare. If his money is mixed (lawful and unlawful), then nothing is incumbent upon you.