Serious problem with Time Machine (external HDD) My Mac Mini (late-2012) got repaired by Apple, and the hard drive was reseted.
Now I want to restore the Mac Mini from my Time Machine Backup (on an external 1TB HDD), but it doesn't appear in the installation or migration assistant.
If I choose to set the Mac Mini up as a new device, I can see the backup in Finder, and it also looks complete (about 700GB for one Backup - pretty close to the storage before the repair). The backup was done completely and also the volume is working just fine. I also updated to 10.9.1 - so the current OS version matches the backed up one.
Any ideas? If I use CMD + R at start and select time machine backup it says there wouldn't be any OSX backups on the backup volume, if I use the migration assistant the "looking for sources" spinner just spinns and nothing happens or is found...

I really need this help because I need both important files from before but also the Applications, it would be really bad and much work for me if I had to restore the files manually - except the fact that the Applications are not shown in Finder...
In the case where you can't restore an entire backup, it's often best to just skip the entire migration and get a few files by hand. 

That often gets you a feel for whether the new OS is working properly or a second hardware issue cropped up or the backup has any issues.

The migration assistant can bring an entire user over once you have some confidence in the various parts of the system.