Doctor Princess Above, War King Below

Two sisters of a noble house: the eldest betrothed to a prince, the youngest to a young lord. When the prince’s illness flares and he nearly dies, the eldest climbs into the crown prince’s bed and becomes his consort, humiliating the prince and wrecking her sister’s life. A modern doctor transmigrates into that unlucky younger sister and is dragged into the prince’s mansion. He’s an invalid, she’s a doctor: he helps clear her name, she cures his illness!