Tighten Terms rather than evict?

[Jeff M.]

I have a tenant who violated the pet policy.  I believe I have grounds to evict but I have never done so before and my lease is not very clear on terms for cancellation/eviction.    The tenant, a young college girl with good reference (I checked before she moved in) apologizes and wants to pay fines and stay through end of lease (5 more months).  I see this is an opportunity to tighten lease, even have her voluntarily cancel lease and sign that she is staying on a day to day agreement.  It seems to me that if she does violate anything in the future, I will have a better lease with clearer rights to evict established.  

I realize it may be best to just pursue eviction or cash for keys now, but I wonder if any of you have had success using an opportunity such as this to establish stricter terms and more managemet control on a tenant.  Thanks.