If the following suggestions work, please let me know and you can Accept the answer (and if possible, leave positive feedback). If not, please DO NOT Accept the answer yet. Instead, let me know and we'll try some more things or I can refer you to another Expert who may be able to help you further.
Without the Mac OS X Install Disk, you may not be able to repair the damage that may be present. In order to proceed you may have to have the Mac repaired or purchase a new Mac OS X install disk.
In the meantime, I would suggest you run some general maintenance on the computer. It is an obvious square-one approach. I normally suggest this first because it is good practice plus if we dig too deep into a problem and simple measures could have repaired it in the beginning we'll be kicking ourselves later... This is going to be a long post so I suggest printing it out for your reference. Also, it will require a USB keyboard, not a wireless keyboard.
Try this, then let me know how things are working out. Make sure you then backup your computer (I suggested Carbon Copy Cloner earlier). After that is complete, then I would attempt to update the system again. This time however, I would update to 10.5.6 (not 10.5.8). IF that works, then move on to 10.5.8. I'll provide those links at the bottom.
First let's reset the PRAM on the computer.
1) Shut down the computer.
2) While the computer is shut down, locate these keys: command (has an apple on it), option (sometimes called alt), P, and R.
3) Turn the computer on, then press and hold Command-Option-P-R before the gray screen appears.
4) Hold these keys down until you hear the startup sound a total of 3 times.
5) Release the keys.
Next, let's try a Safe Boot. This will perform several clean up procedures that may be affecting the machine's performance and startup as well as clear out any cache here as well.
1) Shut down your computer completely.
2) Start up the computer and immediately after the startup chime, press and hold the Shift key
3) Once the gray Apple and progress indicator appear, release the Shift key
4) Note: Startup will likely take substantially longer than normal (20 - 30 minutes is not unheard of)
5) In Mac OS 10.4 or later, you will have to login to your account, even if it normally does so automatically.
6) Shut down the computer.
Now let's check / repair the disk permissions. Although this doesn't fix all problems it is a great way to start when working on a machine, besides it often cleans up any other smaller problems you may or may not be experiencing.
1) Restart the machine.
2) Before you have opened any programs, go to Applications / Utilities / Disk Utility.
3) Click the "First Aid" tab at the top and run the "Repair Permissions" after selecting your Hard Drive from the list on the left.
4) Shut down the computer.
5) Restart and make sure you have everything important backed up.
After you have everything backed up, the next step would be try the 10.5.6 Combo update. Don't attempt the "Software Update" in the apple menu. We'll do it via apple's website instead. I'll provide the link below:http://support.apple.com/downloads/Mac_OS_X_10-5-6_Combo_Update
If you are successful with 10.5.6, then let's upgrade to 10.5.8 from here:http://support.apple.com/downloads/Mac_OS_X_10_5_8_Combo_Update
If you are unable to make it this far successfully, then it may be necessary to purchase a Mac OS X Install Disk for Leopard. If this occurrs, let me know and I'll direct you in the right direction. Good LuckIf I helped,
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