Can a woman shave her head so that it will grow strong again after she had been ill?

Firstly: 

We ask Allah to heal your wife and make her well, and to bless you both with patience and help you to seek reward. 

Secondly: 

It is not permissible for a woman to shave her head unless she has an excuse. 

It says in Fatawa al-Lajnah al-Daa’imah (5/179): It is not permissible for a woman to shave her head except in cases of necessity, because of the report narrated by al-Tirmidhi and al-Nasaa’i from ‘Ali (may Allah be pleased with him), that the Prophet (peace and blessings of Allah be upon him) forbade women to shave their heads. And al-Khallaal narrated with his isnaad from Qataadah that ‘Ikrimah said: The Prophet (peace and blessings of Allah be upon him) forbade women to shave their heads. And al-Hasan said: It is mutilation. Al-Athram said: I heard Abu ‘Abd-Allah (i.e., Imam Ahmad) being asked about a woman who is unable to look after her hair and can she cut short? He said: Why would she cut it short? It was said to him: She is not able to take care of it, and it gets infested with lice and things. He said: If it is a matter of necessity, I hope that there is nothing wrong with it. End quote. 

What you have mentioned in your question is regarded as an excuse, so there is nothing wrong with her shaving her head so that the hair will grow back stronger and thicker, because this comes under the heading of putting things back as they were, and dealing with a fault that has occurred with the hair due to sickness.  

And Allah knows best.