Is the husband entitled to keep his wife's possessions and gifts in her stated health condition, and is he permitted to marry another woman while keeping those possessions to spend on his wife's treatment, especially given his unwillingness to divorce her and his fear of her family's financial demands?

You have done well in preserving the marital bond and adhering to your wife despite her illness and your concern for her well-being. It is not the right of your wife's family to prevent you from marrying a second wife, especially given your pressing need for it and your not falling short in your duties towards your first wife. Islam has permitted polygyny, and it is not obligatory for a husband to seek the permission of his first wife; therefore, it is even more so that he does not need the permission of her family.

Regarding your wife's assets, a person who has lost memory is in the legal position of an insane person. Their ownership of their assets continues, and it is not permissible to transgress against them. A guardian must be appointed to manage her assets, preserve them for her, and spend from them on her. As long as your wife is living with her family and the dispute over these assets persists between you, it appears that her father is the guardian of her assets in this situation. Therefore, you should hand over her money and jewelry only to her father, documenting this with witnesses and writing a receipt for their reception of the money. They are not entitled to demand anything from your personal jewelry or money. However, if you both agree on something you can afford that would be considered part of her expenses, then that is good.