An orphan girl who aspired to be a doctor fell down a flight of stairs after finding out she passed her college exams and was reborn in the body of a noble daughter that was imprisoned in a cell in the dungeons. She helps a large group of beast men and a dragon get free and with the dragons help heads to the neighboring kingdom where both she and the dragon who she named Al are adopted by the prime minister
https://www.shinsori.com/mezametara-chikashitsu/ Funny that both of your LF stories are translated by the same group.