buying with tenants in place and behind (but not on rent)

[Don Griffith]

I am buying a duplex that the owner has been (mis)managing personally.  I think they just got old and started losing interest.  Both sides are really under market for rent, but one moreso.

The really cheap rent of the tenants has been there for 5 years, but never got a rent increase and still owes a balance on the security deposit.  She also has two large dogs that are supposed to be outside only per the lease, but she made no effort to hide the fact that they are inside dogs.  They are well behaved and don't tear anything up so I don't really care about that.  BUT, she was supposed to make a $300 pet deposit when she moved in (5 years ago) and never did.

The leases were just renewed a couple of months ago so I am looking at a year of honoring them as they sit.  How do I honor the leases and still try to secure these deposits?  I thought of structuring an addition monthly payment until the balance is paid.  This would also get her used to paying closer to market rent for the future increase.  I would like to keep her, and she is not going anywhere with 2 big dogs cheap.

ALL ideas and insight are greatly appreciated