If you have already 'talked' to the landlord about the problem, it's now time for Step 2. Send him a certified letter WITH a return receipt. State in your letter that you and he/she have already talked about the problem on such-and-such a date and are still waiting for the repair(s). Also tell him, that if the problem gets worst (such as it starts to drip), you will first attempt to notify him IMMEDIATELY, and if his response is NOT immediate, you'll call a Plumber to fix the problem and deduct the bill from future rent(s). In short, you need to inform him of the 'worstening problem' and give him the opportunity to fix it FIRST. (This is why you must write a letter and notify him FIRST of the worstening problem.) If there's more damage or potential damage to personal belongings (which there is because your 'stuff' is all over the appartment, and you don't know the source or extent of the leak), it needs to be fixed right away.
Do understand that you're a renter and he can opt to NOT renew your lease if you doesn't like your 'attitude'. Personally, I don't think it's unreasonable to put yourself in a defensive position. Any judge will see that you took necessary and resonable measures to minimize damage to the building and your personal belongings. At the same time, if you don't have Renter's Insurance, get it! Get it anyway! If there was a fire, HIS insurance would only cover the building; NOT your personal belongings. At worst, if you're on a month-to-month lease, consider moving. Keep a 'log book' of whatever correspondence you and your landlord and your plumber have. Date and summarize your conversations AND letters. This way, IF you end up in court, you'll be the one that's prepared. (Be sure to start you log book with your initial conversation(s) w/your landlord ...)
My best to ya and hope this helps.
PS: Don't attempt to fix the damage yourself. Pay the pro and deduct it from the rent AFTER the problem worstens AND you've told him of your 'plans' ...