Is it permissible for a man to secretly marry the trustworthy maid to avert temptation, especially since his wife is ill and admitted to the hospital, and he heard that her sponsor is her guardian?

Household servants are not slaves, but rather special employees. The sponsor has no guardianship over a free female servant, as her guardians are her family. It is not permissible for someone who has contracted with a female servant to look at her or be alone with her, because she is a non-mahram. Thinking about marrying a female servant is a sound thought to avoid what is unlawful, but her full rights must be given to her, the marriage must be with the consent of her guardian, the marriage must be announced, and she must not be deprived of her rights such as enjoyment and procreation, while respecting her as a wife.