Is it permissible to speak with a non-devout girl and invite her [to Islam] and offer her tapes with the aim of future marriage, and how should this invitation be, and what are the appropriate tapes?

If you are able to empower some women from among your mahram relatives, or your brothers’ wives, or your friends’ wives, or their mahram relatives, to invite her to commit to righteousness and uprightness, then do so. This is the best and safest approach. There is no harm in sending beneficial booklets, tapes, and brochures to her. But beware of Satan luring you into directly speaking with her, even under the pretext of inviting her to God.