I believe the Florida Record for Number of Hurricanes in a 6 weeks time period would be in 2004. Looks like 4 strikes to the Florida Peninsula in 2004. And 1 strike to the Panhandle.
Of the 5 strikes that year Charley, Ivan, and Jeanne were Major Hurricanes at Landfall. 3 - 13 August 2004 Trop. Storm Bonnie http://www.nhc.noaa.gov/2004bonnie.shtml?