Inherited Tenant Paying $250/month inc Utilities out of lease

[Heather O.]

In a Landlord Friendly County and State, Florida, where there or no rent raise limits, I have purchased a multifamily property with a tenant paying only $250/ month and she has not signed a lease renewal, has a growing list of complaints from before I came into the picture, hasn't allowed my inside as of yet and is consistently late paying.   Until now, she hasn't allowed access and keeps windows covered completely.

What would you do?

(side question - why do tenants assume it's okay to mail a payment on its due date???  We aren't the IRS/CRA!?!)

Have FINALLY set an appointment after 4 months of attempting to, must make a decision soon.  I want a new lease signed.  

Two ideas I thought of or please post your own:

1. Would you step up the rent monthly by agreement??   

OR 

2. Would you give 4 months notice that we are going to HUD FMR rate of $769??

OR 

3. Your idea??

(FYI the identical unit next door (also not updated) I have just rented for $975 to a super clean, reliable tenant)?