Your description reminds me of I Am the Real One.
From Baka-Updates:
A prophecy foretold that the Grand Duke would only have one water elementalist born from him. Though she lacks her father's affection, Keira knows she’s destined for those powers. But after years of doing what was right and proper of a noble, she's executed after the conniving Cosette appears, claiming to be the Grand Duke's real daughter. But then—Keira wakes up with newfound purpose: make changes, allies, and even friends before it's too late. Will it be enough before Cosette hatches her plot?
The day the Grand Duke, Ludwig Parvis, was granted his title, the Oracle prophesised that he'd have a single daughter; a saintess blessed by the water spirits with powers that would save the continent from a crisis.
His firstborn daughter, Kiara, has dark hair, and for the first 18 years of her life, she was his only known daughter, so it was presumed that she was the daughter spoken of in the prophecy, although she'd yet to manifest those powers. She also had a blond, younger brother, Zik.
When Kiara was 18, a girl named Cosette Weinberg turned up, claiming to be the Duke's eldest daughter by a previous wife. While the Duke was unsure of this, he decided to give her a chance to prove herself and allowed her to move in.
Over the next two years, Cosette manipulated things to make Kiara look bad in the eyes of the Duke and his staff. It didn't help that Kiara was so intent on behaving like a proper lady that she came off as being aloof and austere in the eyes of her subordinates.
She became increasingly desperate to manifest the prophesied powers in order to win her father's favour, but her prayers went unanswered, and Colette appeared to manifest the powers instead. As a result, Kiara was branded a fraud and imprisoned, then executed.
She subsequently woke up in bed, two months prior to Cosette's arrival. Having been given another chance, she decided to go about things differently, focusing less on attempting to win her father's favour, and more on becoming friendlier with her subordinates.
One of the first big changes she planned to make was to resign from her position as captain of the knights, and promote the vice-captain to that position instead, since she felt she'd only been given the position due to nepotism, and that he was far more worthy of it.