Operating system running slow, no spyware, malware, virus found.

You didn't mention what processor speed you have, available memory, etc.
Without knowing that, my first suggestions would be to run checkdisk
Press START, RUN type CMD and in console type:
CHKDSK C: /F /X
(where C: is disk drive you want to check...)
/F Fixes errors on the disk.
/X Forces the volume to dismount first if necessary.
You will need your XP cd if errors are encountered.
I would also suggest that you might need to defrag.
Start/Programs/Accessories/System Tools/Disk Defragmenter
Check your Processes:
Open up Task Manager
Select the Processes tab
Click Memory Usage
Take note of which processes are using the most RAM.
David

P4 2.5ghz, 248mb RAM

The computer is an HP, not sure what the model number is.

Process Explorer for Windows v10.21
http://www.microsoft.com/technet/sysinternals/ProcessesAndThreads/ProcessExplorer.mspx

Closely monitor what is using your CPU, and play with the reporting in the app, as to whats displayed. When you find something eating assive amounts of CPU time, then double click it to view further information. Post back any thoughts.....

System Idle Process is using 95+ of the cpu.
Everything else looks normal.

Ok, then check inside the PC for clogged Fans, Fan temperatures etc...May be a heat issue Also, take a look for any bad Capacitors, to indicate failing hardware.

For examples, http://www.badcaps.net

The capacitors were ok, and the fans were dirty but weren't clogged.

The cpu is about 32 degrees celcius.  Don't know if that helps