Restoring Snow Leopard from Time Machine <>

I have a late 2006 MBP that was running Snow Leopard just wonderfully. Yesterday my hard drive failed. I got the question mark folder so I removed the internal drive and docked it to another Mac to confirm it had been fried. I've since put in a new drive (HFS+ formatted with the GUID setting enabled on the partition). I have been running Time Machine with backups to an external drive that is still okay.

Here's the problem: I received Leopard and Snow Leopard from my brother who purchased the multiple-computer license for each OS. I could not find a way to restore the Time Machine backup to a blank drive. It seems that you need to have the OS installed first (right?). So I only have the disks that were included with my laptop, which is like 10.4 (!). 10.4 doesn't even support time machine!

Is there any way to restore my old computer? My brother lives on the other side of the country now, so I'd have to wait a week to get the disks sent. Please tell me there is an easier way!