Is the aforementioned supplication for softening a wife's heart effective, or is there another supplication for getting a wife back?

There is no specific supplication (du'a) for bringing a wife back or softening her heart, but supplication in general is beneficial. It is advisable to involve wise individuals from relatives or righteous people to reconcile between the spouses. If reconciliation is not possible and the wife insists on separation, then separation through divorce (talaq (definition)) or annulment (khul') might be better, as Allah Almighty says: "But if they separate [by divorce], Allah will enrich each of them from His abundance." And divorce can sometimes be a blessing, a release from a burden and a restriction, as Ibn al-Qayyim mentioned; not every divorce is a misfortune.