@HOOfan_1 They don't have a choice. The condition was that they'd be saved from debt if they sent their eldest daughter to be the duke's bride. That was likely the only condition, and homelessness as well as a horrible life of suffering was the only alternative. It's clear their mother(and father, I'm sure) doesn't want it to happen, but as Astina said the father would likely die due to stress and hard labor while the rest of them would likely become prostitutes or mistresses with no power, dying after years of abuse and any STDs they'd likely contract. On one side it's suffering and then death for the entire family, on the other it's "possible" death for one member. The mother can't go cause she's already married, and Kanna was chosen because she was the eldest. It's a shit situation, but I do not envy those parents that had to make such a tough decision they did not want to make.
Also, if it was just a father with their single child, I'm sure the father would have just accepted poverty and worked themselves to death while their child was safe and free from such horrid work as prostitution long enough for her to find a commoner boy to marry, but this is four mouths to feed that the father wouldn't be able to maintain on his own. This means all three would be forced to pitch in during an age where the only working women were prostitutes or women who were already married to working men(farmer's wife, carpenter's wife, etc) and were expected to help them with their work to a limited degree. There were no office ladies, or female blacksmiths, or anything of the sort(except in very,
very rare and unlikely circumstances). If you were a woman you were either a prostitute or a working man's wife, otherwise you'd be dead in some ditch. There was no "independent woman working a full time, office job."