How does Jacob get cured in the end?

Malvina is within the walls the whole time, and when Verena is struck by the same illness that Malvina died of, you see her go through a dream-like fever. In that state, you see her in a tomb next to Malvina, where she runs her hand over Verena's body, in essence placing her soul into Verena.

In the second half of the movie, you see Verena's persona shift as if she was becoming Malvina, which is exactly what was happening. The tomb incident was what "sealed the deal" per se; she told Jakob before she died that another lady will come along and he would know when to speak. Then she ended it by telling him that she was his only mother. When Malvina came back to life encompassing Verena's body, Jakob recognized it in her and chose to speak, saying he missed his mother which Malvina/Verena replied with

I'm here

At the very end, you see her playing the piano. This is because it's Malvina's spirit; you can't just learn to play like that hence why it ended with the recap of their hands: the same hands that were shown in his flashbacks of his mother.

The way I see it - Malvina took Verena's body (cause remember she said she couldn't play the piano and then you see her playing it like a pro at the very end) while she was ill, so that's why she kept saying that another woman will come and that he would speak when the time was right; he was waiting for his mother to come back.

Malvina did not take over Verena's body. Before Verena got sick, she asked the boy to talk so the father would not send her away. She explained that his mother is trapped, and him talking would set her free. After Verena got healed, and the boy talked, they closed and fixed the walls, so Malvina is at peace. Verena learned to play piano because that brings connection between her and the boy (as the boy and his late mother). She is now playing the mother of the boy and the wife of the man.

Malvina didnt possess verina, not entirely, but she became a part of her. In the end she says that she can hardly remember a time before she came here. Shes still concious of herself as Verina. When Malvina is dying she says to Jacob "I am here, i am right here", the same said by Verina in the end when she replies to jacob confiding in her that he misses his mother and is answered with same words spken by his mother "I am here, I am right here". They (M&V) made a spiritual connection that Verina welcomed also. They both ultimately got what they longed for. Verina, A family, and Malvina HER family. Even there names sound like a combination that would include Verina.

I'm thinking that maybe Lilia was the spirit of the cat within the walls (the same cat that was supposedly sealed alive as the building was created, as a sort of guardian or rather a curse) and payed both Malvina och Verena a visit which possibly is the core of the illness rather than a common cold or other (because I get the sense that their illness was under strange circumstances; to say the least). Simply put; Lilia's body expired and the spirit of the cat looked towards Malvina as a new host however died in the process and was yet trapped within the walls. Verena's awaited arrival was their chance (perhaps it needs to be a woman) and at the ending of the film the cat-spirit had yet again gained a consciousness and a body with both the memories and the skills of Lilia, Malvina and Verena (as their souls are either clashed or the souls of the women have passed or, the darkest thought, they are trapped. I could be completely off! I just think that the cat is an important element of the story, cats are often considered sacred and in stories they are often magical beings.

(Edit from comment by OP): Why Jakob was "cured" and could suddenly speak? "Let the words you speak be the words that call me back to you" was his mother's last phrase. That sounds like a spell to me, perhaps Jakob was not possessed but under a spell? When Verena awakens from her illness she at once asks for Jakob. He was like the "spell-carrier" or something.

Was Jakob cured?

The whole film purposes a lot questions and is full of supposition about the natire of Villa and the stones themselves, most of which are left for the viewer to interpret, except for the idea that Verena's body was able house the spirit or consciousness of Malvina and transforms into her by the film's end.

Ghost of Lilia: He will not speak to please you. The boy will speak when the time is right. He's waiting.

In terms if Jakob is in fact sick or cursed, it's not entirely clear, but one line of dialogue does suggest that Jakob is intentionally waiting for the right moment to speak, meaning that there is some kind of supernatural or metaphysical semantics at work.

Malvina (to Jakob): Another woman will come. You will see. There will be another. You will know her by the love she shows to you. Let the words you speak be the words that call me back to you. Promise me. Promise.

But viewers can easily get the gist that voices of deceased, and especially Malvina's, can be heard throughout the stones in and of the villa and throughout it's grounds acting as conduit, medium, or nexus and all seems to working towards Jakob getting his mother back through conditioning Verena, something that Malvina told Jakob could happen before she died.

Could it be perhaps, that Verena was Malvina the whole time and that, in the end, she finally broke free from the illness that had held her the whole time?

Did she, Malvina, go mad due to the fever and "live" a dual life in her head before she finally came to her senses?