Brother's wife is refusing to sign divorce papers. What can he do?

Great question.  This actually happens quite a lot, where one party may be the party filing for divorce and the other won't sign for one reason or another.  In Texas, Sec. 6.701.  FAILURE TO ANSWER.  In a suit for divorce, the petition may not be taken as confessed if the respondent does not file an answer.   Added by Acts 1997, 75th Leg., ch. 7, Sec. 1, eff. April 17, 1997. Sec. 6.702.  WAITING PERIOD.  (a)  The court may not grant a divorce before the 60th day after the date the suit was filed. A decree rendered in violation of this subsection is not subject to collateral attack. (b)  A waiting period is not required before a court may grant an annulment or declare a marriage void other than as required in civil cases generally. You can divorce regardless of the other person's actions.  Oftentimes, you may have to go through the process of appearing in court and having the judge simply enter the order.

You can get a "divorce by default".  You will need to have your papers prepared and file them with the court.  After filing, a citation will be issued and your spouse served.  Then, if after 20 days your spouse does not answer (ie they default) then you will file your final paperwork and appear before the judge on or after the 61st day.  A divorce will then be granted on the terms that you requested.