Which external...E: or F
Actually should be on both in exact same folder on each. If this is true then everything (except Hoster database) should be ok. If the files don't show up on F: then your copy didn't work.
I just had a light bulb flash!!
Is this "My book (E) SC8517-4.kma" the whole file name for one song?
The "(E)" being for English and I suppose an "(S)" for Spanish? If this is true how did you go about altering the names of the files? When you import CDG discs they are named with the DiscID-Track by Hoster.
I don't think that should cause a problem but it could be.
If you find the files on the F: drive through Windows Explorer then open Hoster and rebuild the database pointing only to the F: drive. Then try playing one. If it works then you should be able to rebuild the database again with the E: drive only. Then you should be able to remove the F: drive and move it to the other computer. Note that the drive letter may not be the same on the other computer, so when you build the database use the new drive letter.