Will the one who introduces the internet into his home be sinful if one of his female relatives uses it in a way that displeases Allah, given his prior knowledge of her misuse, and then his forgetfulness of that when reintroducing it? And is this considered an ongoing sin? What is the solution, given the difficulty of preventing her?

Since you have learned that your relative is using the internet for impermissible matters, you are not permitted to enable her to do so. Otherwise, you would become a partner with her in sin, as Allah Almighty says: ﴿And cooperate not in sin and aggression.﴾ Just as the one who points to good is like the one who does it, so too is the one who points to evil like the one who does it.

You must admonish her with a good approach and remind her of Allah and the obligation to thank Him for His blessings by using them only in what pleases Him. If she does not respond, then you must prevent her from using the internet. Seek Allah's help and remember the Prophet's (peace be upon him) saying: "Whoever seeks Allah's pleasure despite people's displeasure, Allah will suffice him against the people. And whoever seeks people's pleasure despite Allah's displeasure, Allah will entrust him to the people." And pray for her guidance.