I suggest you stop using the Migrastion Assistant. Simple fact is, as you have found out, that it does not work most of the time and when it fdoes actuall transfer data it doesn't do it very well.
Just network the 2 computer together over your LAN and copy and past the files you want to move to the Mac.
Otr use an External drive, formatted either FAT32 or exFAT (doinig the formatting on the Windows PC) and then copy to the external from the Win PC and then copy from the external to the Mac.
You'll be happier.
. . .
But then I finally thought I managed to get it working. It said that process was complete (on both the PC and mac), but there was no data on my computer and no music in my iTunes.
Your data is in a new user account created by the migration.
When you next log on, you'll be prompted to create a password for it.
See the last few lines in OS X Lion: About Windows Migration Assistant