Is it permissible in Islamic law to speak with a girl to inform her of a proposal to marry her, or to encourage her to convince her family to agree to the marriage, knowing that the conversation is limited to matters of marriage and their future life together?

Speaking between a man and a woman is not permissible unless there is a need, and its permissibility is conditioned upon: safety from temptation (fitna), not speaking in a soft and alluring tone, and the conversation being limited to what is necessary or needed. As long as the conversation between you and the girl was within this framework, there is no harm in it. You should try to propose to her again through her guardians or intermediaries, or inform her of your desire to propose, while limiting the conversation to that. If the proposal is not accepted, then ask Allah to grant you someone better than her.