Is not informing the manager that the employee was the one who did the work considered a mistake, based on the Almighty's saying: "We wish no reward from you, nor thanks," and should she correct that, or continue working for the sake of Allah?

The employee entrusted with a task is obligated to perform it. They have the option to inform their manager that they are the one who performed it, unless failure to inform would result in harm. The general principle is that management is responsible for monitoring employees and their work. It is commendable to inform in order to preserve rights. It is necessary to inform officials if it is known that an employee has lied and claimed to have performed work that they did not do, in order to prevent wrongdoing and preserve rights.