How many files do you have to transfer? I.e. what is the total amount of data in Kb or Mb or even Gb?
I presume its not small enough to warrant compressing the files onto floppy disks and effect the transfer that way. Nor do you have access to an external CD-writer. Nor a network access card so you could connect to a network which has a PC with a CD-writer already installed.
It would be also helpful if you provide more details of your Toshiba laptop's specification even if you do not yet know the new laptops specification yet.
You you get really stuck, you could always go to a PC World with a service department, or a local PC shop who could transfer your data onto CD-Rs for you to copy onto your new laptop, which presumably will have a CD-drive at least?