I'm not sure what a statement from your previous landlord has to do with the issue unless the previous landlord owned the property you are having the problems with. The landlord is responsible for extermination unless your lease says otherwise. You would need to send the landlord a certified letter advising them of the problem and giving them seven days to cure the issue.
You should consult with a local landlord/tenant attorney for a consultation to see if you have a case.
This communication is not intended to create an attorney/client relationship. It is always recommended you consult an attorney in person to discuss your case. The Law Offices of Stage & Associates practices state-wide and represents homeowners and community associations. Please visit our website at www.stagelaw.com.
Unfortunately, landlord/tenant law is an area that is governed by both statutory and contractual law, as well as decisions handed down by appellate courts. Thus, they are very fact specific to your particular matter.
First, you must look to the contract and the obligations/remedies that the contract provides for both parties. If your contract references who is responsible for exterminations, then the contract will govern your particular fact pattern. If it does not, it is likely that a court would rule according Florida Statute 83.51(2)(a), which states that "the landlord of a dwelling unit other than a single-family home or duplex shall, at all times during the tenancy, make reasonable provisions for:
1. The extermination of rats, mice, roaches, ants, wood-destroying organisms, and bedbugs. "
You should retain an attorney to discuss your situation, as Florida statutes and your contract will determine how you respond to your landlord's failure to comply. There are statutes that specifically deal with the procedure in which you must inform the landlord, as well as time frames that the landlord has to comply after that notice is provided.
The information contained in this answer is not,nor is it intended to be, legal advice. Please consult an attorney for information related to your specific situation. Further, this answer does not form an attorney-client relationship between the attorney and the individual(s) or entity requesting information.