Help! My Time Machine Volume is the only one with the files I need!

I have been nursing along a 2002 (as I recall) G5, and I think it's brains are just worn out. (I primarily use my MBP Laptop, now)

I have a 2TB inside it as the Time Machine back-up, and recently was getting too many "beachball of death" cursors on the startup drive (inside), so I did a final backup and reformatted the Start-up drive and reinstalled 10.5, upgrading to 10.5.8 via internet.

I have attempted several times to Migrate from Time Machine my settings and files, but it craps out in the process — giving it hours, even days to complete.

Many, many photos and music files, videos and recordings used to live on that Startup drive, and are all backed up to the Time Machine drive.

Is there a process, even a techie one, for gaining permissions to recover my files to a new, blank drive... or even to my MBP selectively? I am getting permission warnings and all — and I am the only one who has ever used either computer!

Can anyone help me?

—David