Migration assistant using backup folder already on computer?

I had my computer repaired which included a new logic board and hard drive. Before sending it out, I had my hard drive backed up to an external drive. During repairs, somehow the connecter on the external failed, and I had to pay to have to backup moved back over to my computer.

Now I'm sure sure how to reimplement all my data properly. Right now there's just a folder on my desktop called Backup that has all my old hard drive data on it. Can I use this folder with migration assistant or do I have to manually find where everything should go and drag it to my new hard drive?