His wife and her family accused him of having something wrong with his mind; should he take a second wife?

Proof that a person is in good mental health and treats others well does not require the testimony of a doctor. What matters is what people see of a man, not the papers he produces.  

Hence if the accusations of your wife’s family are based on things that they have seen you do or words that they have heard you speak, then they may have a reason. You have to sort yourself out and mend your ways so that there will be nothing for others to judge you by.  

If what they said about you has no real basis, rather it is false and sinful testimony, then we think that you should advise them and explain to them that their false accusation against you is sinful, and tell them how that has caused division between you and your wife. If they recant, then all well and good, otherwise you should shun them, and not let them visit you or let your wife visit them, lest they turn her against you. 

With regard to your intention of taking a second wife, what we advise you is to wait and to examine the real motive for that, because often in such cases the decision is based on the desire for revenge against the first wife and her family because of the trouble they caused, and usually when a man marries with this attitude it leads to trouble for the second wife too, if the husband has not set things straight with his first wife and her family. 

If you have not set things straight between you and your wife and her family, we think that if you want to take a second wife, you should let the first wife go and divorce her, or you can keep the first wife if you intend to keep both wives if Allaah guides her (the first wife). We also advise you to have a good intention and try to do the right thing, for Allaah says concerning spouses between whom there are difficulties (interpretation of the meaning): 

“if they both wish for peace, Allaah will cause their reconciliation. Indeed Allaah is Ever All‑Knower, Well‑Acquainted with all things”

[al-Nisa’ 4:35]

We advise you to fear Allaah with regard to every step you take and not to let this problem cause you to treat her unjustly or to keep reminding her of mistakes that she made but that she has apologized for.

 We ask Allaah to set your affairs straight and to make it easy for both of you to do the right thing. And Allaah is the Source of strength.