What rights do I have if landlord refuses to fix a shower leak?

Google your state's landlord/tenant laws. I have yet to Google a particular state and not find a paper written by an attorney that explains what you need to do.

In general, you need to start a letter writing campaign. Every state has a procedure for repair and remedy. Google it. In general, you must notify the landlord in writing. I always suggest certified mail with return receipt. It forces the landlord to sign a card that is returned to you so you have proof they received the letter. Your state laws will define how long you have to give your landlord to repair it. In my state, they say "reasonable time". Typically, two weeks suffices here, but other states mandate a time frame. In most states, there are laws regarding fixing leaks. If anything, many of the states are more apt to slap a landlord's hand for not fixing a leak than they are to make them remedy mold. Mold is still an iffy issue in most states. If the landlord still refuses, now you have proof through letters of you trying to get him/her to fix it. At this point, some states let you sue, break your lease, deduct the repair from your rent, or withhold rent. Again. Google your state's tenant/landlord laws and follow their procedures.

Your option is to continue to clean up after the leak. "Mold" (most likely mildew) does NOT render premises unfit for human habitation. Yes, the landlord SHOULD repair, if only to protect the value of the property, but he is not required to do so.

If he continues to refuse to fix it, call the Health Department and let them see it. Mould is poisonous and it can affect your breathing.