Is it permissible to mortgage a plot of land—which I bought with great difficulty for the purpose of building a house—to a usurious bank in order to obtain a sum of money, so as to invest it in a small project that helps meet the household's needs and treat the sick daughter, and then repay the mortgage with interest from the project's profits and retrieve the land?

It is not permissible to take an interest-bearing loan from a bank to establish a project, because it is not a necessity that permits committing what is unlawful. Allah has forbidden Riba (usury/interest) and warned those who consume it of war. The Messenger of Allah, peace and blessings be upon him, cursed the one who consumes Riba, the one who feeds it, its scribe, and its two witnesses. The questioner must be patient and seek lawful means of livelihood, for whoever fears Allah - He will make for him a way out and provide for him from where he does not expect.