Does Islamic law permit an elderly and sick woman who uses a diuretic medication, which makes her need to relieve herself approximately every ten minutes, to pray without performing wudu?

It is not permissible for a Muslim to pray without ablution (wudu') if they are capable of performing it. The mother should relieve herself, perform ablution, and pray; ten minutes are sufficient for this. If separating the prayers is difficult for her due to her illness, then she has a concession, according to many jurists, to combine the prayers. However, abandoning ablution is not permissible as long as she is capable of it. If she is unable to maintain her ablution for a sufficient time to purify herself and pray, then her ruling is like that of a person with incontinence: she should purify herself after the time for the prayer begins, and what exits from her thereafter does not harm her prayer.