Unfortunately using a USB floppy drive hasn't worked, I've tried that before and none of the OSs will mount the drive.
Option 1
Being an older machine, the best option for recovery would be if the Macintosh II has an Ethernet port (10baseT) and you have a machine that supports AppleTalk (10.5 and older).
You can then connect your Macintosh II directly to your network, possibly allowing your current computer to connect to it through AppleTalk. If your machine runs 10.6 or newer, I would suggest picking up a free G3 iMac for this purpose.
Option 2
Pick up an older PCI PPC Macintosh (i.e. 7600) with a serial connector (9 pin DIN). Add a USB PCI Card, and install 9.x
Once done, setup appletalk, and copy the files to the PPC Macintosh (very slow). Then copy files to USB and transfer to your current machine.
Option 3
You will need a PCI SCSI card, a compatible Macintosh (I.e. Power Macintosh G4) and an external SCSI HD case.
Install the Macintosh II HD into the case, and connect to the Power Macintosh G4. Then copy files to USB and transfer to your current machine.
Additional Information
Floppies formatted on 800k like those from your Macintosh II cannot be read by a Macintosh running 10.x
Computers with 10.x support and a internal floppy drive are not an option as the floppy drive is disabled by 10.x
Other Suggestions
Find someone around your area that has all the equipment for you to transfer the files.