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?

9 - 14 August 2004
Hurricane Charley- Rapidly strenghtened to a Cat. 4 at landfall
http://www.nhc.noaa.gov/2004charley.shtml?

25 August - 8 September 2004
Hurricane Frances- Cat. 2 at landfall
http://www.nhc.noaa.gov/2004frances.shtml?

2 - 24 September 2004
Hurricane Ivan-Panhandle Strike
http://www.nhc.noaa.gov/2004ivan.shtml?

13 - 28 September 2004
Hurricane Jeanne Cat. 3 at landfall
http://www.nhc.noaa.gov/2004jeanne.shtml?

http://www.nhc.noaa.gov/tracks/2004atl.gif