Is what this girl did permissible or forbidden, where she collected money to pay off her debt under the pretext that it was for a sick person?

It is not permissible to use money collected to help a sick person to pay off a debt without the donors' permission, because the agent is restricted by the purpose for which he was appointed. The donors can be consulted, and it can be clarified to them that the news about the patient is incorrect. Then, their permission can be sought to spend the money on helping those in debt and distress. If they agree, then it is permissible to use the money to pay off the debt.