All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
Most of the problems that occur between the wife and her mother-in-law are due to suspicion or some misunderstanding about some behaviour. So we advise you to be wise in trying to solve this problem. Try to investigate the reasons, and reconcile your mother and your wife, even by using Tawriyah (saying something which has more than one meaning and intending a meaning different from what the listener is likely to understand). Additionally, Islam has even permitted lying in order to reconcile people as it is confirmed in a narration reported by Imaam Muslim . You can, for instance, tell your mother that your wife respects her and holds her in great esteem, in order to repel hatred that is in the heart of your mother against your wife, and to comfort her about your wife, and so on. You have to advise your wife as well to be patient about your mother as a sign of being kind and dutiful to you, and a means of strengthening your marital relationship. If your wife behaved in a way that your mother misunderstood, then you have to clarify this matter to her, and your wife has to apologise to her if she did not fulfill any of her rights.
You have to know that it is your wife's right on you to provide her with a separate accommodation with its own amenities, even if that is in the same house, and she is not obliged to dwell with your mother. So if you can fulfill this without causing any harm to your mother, then do so.
Finally, we should note that if there is a need to reside in one house, then the woman has to wear Hijab and observe all the religious requirements with the husband's brother, as he is a marriageable man (to her). We also advise your brother to be careful about listening to what could lead to the cutting of kinship, or causing negligence towards keeping the bounds of kinship.
Allaah knows best.