Will the wife be sinful for taking a sum of money that fell from her husband and using it for her treatment due to her lack of money, with her intention to repay it?

You are mistaken in two ways: withholding money from your husband without a legitimate شرعي reason, and disposing of it without his permission, unless there is an urgent necessity for the disposal, in which case there is no harm to the extent of that necessity. You must return the money to him or seek his forgiveness. If you fear harm from his knowing, you can return it indirectly.