When i had DD2, they were happy to let me go for nearly 2 weeks after my water had broken. i had to have AB's, and come in every second day for a b/t and monitoring. so technically u can go awhile afte rur water breaks. I think i read somewhere that if u get PPROM u generally deliver within a week.
But if your waters had been broken by the doctor though i'm not sure because you've agreed to the induction and thats part of the process. But yes u can ask for more time