While it is true that tenants can be held responsible for the cost of clogged drains, it normally is covered by the landlord. My policy is to just keep my tenants happy, even if it cost me a few dollars here or there.

Like your drain, your kind of stuck. Slow running drains are caused by hair, grease, etc. So technically, the problem is due to you living there. Clogged drains can indicate one of two things:

1) Tenant has been pouring inappropriate things down drain such as grease, or excessive food, tampons, etc. (toys if you have kids)
2) Possibly the drain is slow due to a plumbing issue. I have a property that will clog every 6 months due to poor drain design.

You may want to try using chemical options such as draino. If that doesn’t clear it up, then you will probably need to hire a professional. You can still try to negotiate with your landlord on cost, but legally you are up against a wall. Your landlord is within reason to charge you for the cost, even if it is poor business practice.