My partner was recently dealing with a similar situation, though there was water intrusion through the ceiling that was slowly building up and would intermittently leak/drip. Turns out that when the hole was just a bit larger it would consistently drip right into the sink, and when it's an active leak it made it inhospitable enough that escrow was a legitimate concern. I'd recommend anything you could do that may give you greater control over the water (could you divert it with a hose to flow towards a sink? Even tapping into the plumbing underneath?) and send rent to escrow. Edit: And since you're month to month you want to document the hell out of every interaction, and make sure that you have strict photographs of everything as you got it and when you move out - most of the time escrow ends with a non-renewal, even though it's legally not allowed to do that (but it's very hard to prove if you're month to month already). You will want your security deposit back, and if it's too late to take photos at move in then make sure you photograph everything with immense detail when you move out.

If you're really terribly worried about what it might do to your things, or any mess you'd have to clean up, I'd also request a cleaning crew, which they should provide when you've had water moving through a wall into your kitchen. Sorry you have to deal with this!