Not without spending a bunch of money.
When you delete something from a hard drive, the bits are technically still there -- just the "index" that tells the operating system where to find those bits is gone (kind of like taking the table of contents out of a book, then trying to find a specific chapter when all the chapters are mixed up). If you don't use the computer anymore and lessen the chance of writing over those bits with new bits (which would make the data nearly 100% unrecoverable), then you stand a good chance of recovering the data by sending the drive to a hard drive forensics company, like drivesavers.com... but prepare for sticker shock: data recovery costs anywhere from a few hundred dollars to a few thousand dollars.
The easiest way to get the data back is to simply restore it from the Time Machine backups (or any other backup) that you regularly keep (since Time Machine makes it easy and transparent to back your entire computer up without manual intervention at all).
If you don't have a Time Machine backup, shame on you, and it may be time to sit your daughter down and explain to her that daddy had the opportunity to save the data, but consciously chose against doing so because (insert horsehockey reason here).
Are you sure the data is nowhere on the hard drive? Could it be that you created a new user account, and the data is stored under a different user folder? Which precise option did you select when you "reinstalled my mac 10.5 [sic]?" -- "Archive & Install," "Erase and Install," or "Upgrade Install?"
Seriously, drivesavers.com can (potentially) save your bacon. Just be prepared to empty the bank account doing so.
