I have a Toshiba Satellite and the sound quit working after a Windows Update a few months ago. I rolled back the audio driver to the previous rev and it started working again.
Google “USB Sound cards”.
Plenty of devices that will work with Toshiba junk. Glad that yours worked so long for you.
As for the ATX case, yes you can build a new system. ATX is still supported.
There are plenty. The Sound Blaster Extigy was a high end system that was around for nearly a decade, even before USB 2.0.
For basic sound in a laptop, there are a number of compact solutions available at a good price.
Startech actually has a decent reputation for low priced gizmos.
I broke the headphone/line out jack on my old Toshiba Satellite. The onboard speakers sound just awful. I bought a Creative external soundcard for about $30. It is about the size of a deck of cards. It connects via USB and works great- gives me headphones/external speaker connection and stereo in/outputs.
Your ATX case will work fine with just about any of the standard sized boards out there. The only thing that you will need to do is make sure that you get the right sized power supply and cooling setup for whatever hardware you install.
http://www.epinions.com/content_89536106116
This implies that the case can handle an ATX motherboard.
HOWEVER, if yours came with a power supply, that might not work so well with the newer motherboards, especially if you are using a fancy video card.
If the there is a 20/24 pin main connector mismatch, adapters can solve that.
ping

Turtle Beach makes some nice little external USB devices. I’ve got an Audio Advantage Amigo which provides sound in and out, including digital audio out.
It was about $30 two years ago. It is about the size of a USB flash drive and provides good sound.
Personally, unless there is a physical problem with the computer I would suggest that you try backing up your data and reinstalling the OS from scratch before you do anything else. Sound problems are usually driver issues.
Your ATX case will work fine for a new build, but as others have said, you will probably need a new power supply. Besides the 24 pin power supply, SATA drives have their own power connectors and generally current computers have much higher power requirements.
I’ve had a soundblaster USB audio system for years. Works fine, sounds good. It might have cost $30? $40? The speakers were much more.
Bose Companion 5 sound system is USB and does not require a sonud card.
What's the deal with "90's technology"? That sounds like a cop-out to me.
THEY PURPOSEFULLY BREAK THEIR OWN HARDWARE DRIVERS TO FORCE UPGRADES!
Try the Turtle Beach Micro external sound card... at http://www.turtlebeach.com/products/micro/home.aspx
If you shop around you can get it for about $25 delivered w/ no tax. Works great for my circa 1999 Sony laptop.
First off, for the laptop... download this:
http://www.reallyeffective.co.uk/sound-control/
Sound Control
First released in July 1999 and updated regularly since, Sound Control is a freeware replacement Windows Sound Mixer applet which together with most of the features of the standard Windows mixer it also allows you to add hot keys to adjust any combination of your sound mixer volumes. Support to control Winamp with a series of hot keys, CD Player and an On Screen Display is also included. Supports Windows 95 / 98 / NT4 / 2000 / XP and Millennium-Me.
If you want to run the smallest possible program to replace the windows master volume controller.. try this:
http://www.codesector.com/audiosliders.php