How can one act concerning assisting a brother's children with a monthly salary, coupled with intense fear of showing favor or revealing the assistance, especially since the assistance comes after a previous wrongdoing by the brother?

Providing for your needy nephews and nieces is a righteous deed for which you will be rewarded if done seeking the face of Allah. Fear of reminding them of your favors is commendable, but it should not be a reason to stop providing for them. The dispraised form of reminding someone of favors is recounting the blessings bestowed upon the recipient. If you do not do that, then you are not one who reminds of favors, nor will you incur the sin of reminding. Strive to make your provision purely for the sake of Allah, with the intention of drawing closer to Him, maintaining kinship ties, fulfilling their needs, and protecting them from moral degradation. Responding to ill-treatment with kindness is legally commendable. Continue to provide for them and be gladdened by good and reward from Allah.