Is it permissible for my sister to pray without re-performing ablution, given her frequent need to use the restroom and the slight emission of air without a smell or sound, which prolongs her stay in the restroom and hinders her prayer?

If the sister finds enough time to perform the prayer with proper purification, she must wait for it. However, if the bleeding is continuous all the time, her ruling is that of a person with incontinence: she performs ablution after the prayer time begins and prays with that ablution as many obligatory and supererogatory prayers as she wishes until the time for that prayer ends. What exits from her does not harm her, and her ablution is nullified by her voluntary impurity or by the expiry of the prayer time.