Sounds like the floppy diskettes may be bad.
1) If you can get the data from the old computer then copy it to different floppy diskettes. Then use these different floppy diskettes to copy to the new computer.

2) Try doing a diskcopy of the "bad" floppy diskette. Sometimes a diskcopy will be able to copy the disk.

3) Copy all of the files that you can to the hard drive. Which files were not able to be copies? Depending on the file, the data may be able to be recovered or recreated.

4) Run scandisk on the floppy disk. Sometimes it can fix the diskette.

Do you have a second backup for the end of year?