What is the way to seek divorce from my wife, given that the judge refused it due to lack of witnesses, after she lied and swore that she still wished to live with me and wanted financial support, and she is currently residing with her ill-mannered sister?

The fundamental principle is that divorce is in the husband's hand and does not require a judge to effectuate it. A rebellious wife (nashiz) is not entitled to maintenance (nafaqah). Uttering the words of divorce and having two just witnesses suffice for divorce. There is no obligation for maintenance for a disobedient wife who leaves the house without permission. If the judge is a sharia judge, he might have relied on unmentioned matters. If the law places divorce in the hands of the judge, then it is advisable to seek the help of a lawyer to obtain a divorce according to the law. Having witnesses attest or appointing a lawyer in this case is not considered among the marital secrets forbidden to disclose.