Hi - my name is XXXXX XXXXX I'm a Real Estate
litigation attorney in Tennessee. Thanks for using JA! I'll be glad to assist you.
The short answer is NO, you don't have to wait 30 days to remove the tenant if she's delinquent with her rent. Instead, you must give her a 3 day written notice to pay the rent current or vacate, and if she doesn't do either after the 3 day notice period expires, you would then have the right to sue her in general sessions court for possession and for any unpaid rent, late fees, etc. that she has amassed.
Her claims about repairs, etc. are likely unfounded. In order to legally withhold rent in Tennessee, the tenant must give you notice in writing of SERIOUS issues that need repair - - these are usually things that are required to make the property inhabitable. Here's a good article you can read: http://www.nolo.com/legal-encyclopedia/tennessee-tenant-rights-withhold-rent-repair-deduct.html. Since this hasn't happened, her claims that she's withholding rent due to a minor water leak likely isn't legally justified.
Thus, all you should have to do is issue a 3 day notice to pay or quit, and if she doesn't pay up and or vacate, you can file an unlawful detainer action in general sessions court and seek an order of possession and a judgment for any unpaid rent or fees.