Are you in a single-family home or duplex? If so, the answer is that it depends on your lease because Florida law allows owners of single family homes and duplexes to set determine these responsibilities by the lease. The default though is the landlord would have to cover it. If the lease doesn’t mention anything about pest control, then yes you could break your lease because FL statute 83.51 requires the landlord to make reasonable provision for “the extermination of rats, mice, roaches, ants, wood-destroying organisms, and bedbugs.”
If you are in a multi-family dwelling, the landlord has to provide pest control. I would give him a seven-day notice that he must take care of the termites or the lease will be terminated and you will be moving out in accordance with the Florida law.