I have 1.5 months of security deposit.

She has offered to pay 3 of the 5 months  but keep her deposit and last months rent.If we let her go, having some buffer to find a new tenant would be nice but the fact that we need to find one during the holidays and leading into winter does not sit well with me.

Another option is this: what you can do is take a deposit to hold the place (and have a signed deposit to hold agreement), but they forfeit the deposit if they fail to rent.

@Brandon Carlson no deposit? 

Maybe they really want to move into a SFH and they need a larger deposit to do so, you could trade some cash for those keys in that manner....You also mentioned that the rent was below market, if they are on a month to month lease, then you should be able to start systematically raising rent to bring them up to market. 

I uploaded a pet addendum to apartments.com, asking her to fill out, along with a pet deposit.  

I have a set deposit go in every two weeks and have that investing into a mix of stocks and bonds.

Not only that but you can automatically screen tenants with an application and credit and background checks, collect rent through the app that will automatically be deposited into your account, automate late fees, itll send emails to your tenants when their lease needs to be renewed or when there rent is due soon/late, it'll automatically generate profit and loss statements and tax documents and more!

They moved in on November 15th so I have recieved $2,000 (deposit) and $800 (prorated rent). 

I'm assuming you want to grow your portfolio.....don't start off by on the wrong foot with a Housing Authority. 2)   Get as large of a security deposit as possible.