I've been interested in Killing Floor on Steam, so I decided to grab UT2k4, and check out the original mod. When I try to run it though, I get this error:
Any ideas?

Usually thats because of an error in an ini file.
Can you paste the contents of your ini file and your core.system?
Also make sure your directx is is updated.
Try a fresh reinstall as well.
it should be like this
[Core.System]
PurgeCacheDays=30
SavePath=../Save
CachePath=../Cache
CacheExt=.uxx
CacheRecordPath=../System/*.ucl
MusicPath=../Duffers/Music
SpeechPath=../Speech
Paths=../System/*.u
Paths=../Maps/*.ut2
Paths=../Textures/*.utx
Paths=../Sounds/*.uax
Paths=../Music/*.umx
Paths=../StaticMeshes/*.usx
Paths=../Animations/*.ukx
Paths=../Saves/*.uvx
Suppress=DevLoad
Suppress=DevSave
Suppress=DevNetTraffic
Suppress=DevGarbage
Suppress=DevKill
Suppress=DevReplace
Suppress=DevCompile
Suppress=DevBind
Suppress=DevBsp
Suppress=DevNet
Suppress=DevKarma
Suppress=RecordCache
Suppress=MapVoteDebug
Suppress=Init
suppress=MapVote
Suppress=VoiceChat
Suppress=ChatManager
Suppress=Timer
Paths=../Duffers/System/*.u
Paths=../Duffers/Maps/*.ut2
Paths=../Duffers/Textures/*.utx
Paths=../Duffers/Sounds/*.uax
Paths=../Duffers/Music/*.ogg
Paths=../Duffers/StaticMeshes/*.usx
Paths=../Duffers/Animations/*.ukx
;Paths=../Duffers/Saves/*.uvx

I can't seem to locate anything named core.system in any ini file I've been able to find.
Which ini am I looking for, and where should it be?

I think I have had the same problem.
If I remember correctly the cause was the special splash screen that comes with KillingFloor. For some reason later UT2004 builds couldn't handle it anymore.
My solution was simply to replace the KF splash screen with the default UT2004 one. This splash screen is just a bmp file, which I think is located in the Help subdirectory of your UT2004 installation directory. This file is called UT2004Logo.bmp or something like that. Copy it to the Help subdirectory of the KF mod directory. You may have to rename it to the KF logo file name which should be there to make it work.
Sorry I can't be more specific on file names etc., but I have no longer installed UT2004, let alone KillingFloor. I am pretty sure though that if UT2004 is running fine by itself, that this is causing your problem.
Good luck, bj
PS: Be advised that the standalone KillingFloor as found on steam is really a huge upgrade compared to the original mod. The gameplay has not changed really though.

hmmm i've re read your post and have you tried a clean install of the game?
also the things i noted where in the Duffers.ini
there you'll find
[Core.System]
...
[Engine.GameEngine]
...
[WinDrv.WindowsClient]
...
the settings described in Core.system is what you need to post so we can have a looksy

Make sure your UT2004 is latest version (v3369) and that you have ECE content.
If you don't have that, install UT2004 megapack.

I'm having the same problem. I keep getting the "Could not spawn a GUI Controller!" Error each time I try to run Killing floor. I have looked in my System Folder for Killing FLoor and found the Core.System info in the Default.ini file. Here are its contents:
[Engine.Engine]
RenderDevice=D3D9Drv.D3D9RenderDevice
;RenderDevice=D3DDrv.D3DRenderDevice
;RenderDevice=Engine.NullRenderDevice
;RenderDevice=OpenGLDrv.OpenGLRenderDevice
;RenderDevice=PixoDrv.PixoRenderDevice
AudioDevice=ALAudio.ALAudioSubsystem
NetworkDevice=IpDrv.TcpNetDriver
DemoRecordingDevice=Engine.DemoRecDriver
Console=KFMod.KFConsole
GUIController=KFGUI.KFGUIController
StreamPlayer=Engine.StreamInteraction
Language=int
Product=KillingFloor
GameEngine=Engine.GameEngine
EditorEngine=Editor.EditorEngine
DefaultGame=KFMod.KFGameType
DefaultServerGame=KFMod.KFGameType
ViewportManager=WinDrv.WindowsClient
;ViewportManager=SDLDrv.SDLClient
Render=Render.Render
Input=Engine.Input
Canvas=Engine.Canvas
DetectedVideoMemory=0
ServerReadsStdin=False
[Core.System]
PurgeCacheDays=30
SavePath=../Save
CachePath=../Cache
CacheExt=.uxx
CacheRecordPath=../System/*.ucl
MusicPath=../Music
SpeechPath=../Speech
Paths=../System/*.u
Paths=../Maps/*.rom
Paths=../Textures/*.utx
Paths=../Sounds/*.uax
Paths=../Music/*.umx
Paths=../StaticMeshes/*.usx
Paths=../Animations/*.ukx
Paths=../Saves/*.uvx
Suppress=DevLoad
Suppress=DevSave
Suppress=DevNetTraffic
Suppress=DevGarbage
Suppress=DevKill
Suppress=DevReplace
Suppress=DevCompile
Suppress=DevBind
Suppress=DevBsp
Suppress=DevNet
Suppress=DevLIPSinc
Suppress=DevKarma
Suppress=RecordCache
Suppress=MapVoteDebug
Suppress=Init
suppress=MapVote
;Temporary
Suppress=VoiceChat
Suppress=ChatManager
Suppress=Timer
Paths=../Textures/Old2k4/*.utx
Paths=../Sounds/Old2k4/*.uax
Paths=../Music/Old2k4/*.umx
Paths=../StaticMeshes/Old2k4/*.usx
Paths=../Animations/Old2k4/*.ukx
Paths=../KarmaData/Old2k4/*.ka
Does that tell you anything?

Next time just delete your game's INI files - UT2004/user.ini and KFMod20/KFMod20user.ini.