My Inspiron 6000 speakers stopped working but headphones work OK

So am I just dealing with a dead hardware sound amp on the motherboard? 

It could be mechanical failure in the headphone jack, where the contacts are failing to mate after removing the plug. Try gently inserting and removing a q-tip. Could also be a loose connection in the palm rest assembly. Consult the Service Manual for disassembly instructions.

How do I run the extended sound diagnostics?

To run Dell Diagnostics

1. Restart the computer and tap the f12 key as soon as it starts to boot up .
2. Select 'Diagnostic' from the boot menu. The computer will run its pre-boot assessment tests (PSA) before Dell Diagnostics starts. During the PSA you will hear some tones through the system speaker (but not through the headphone jack) but this does not adequately test the audio hardware. Dell Diagnostics plays an instrumental piece of music complete with drums, so if you only hear some tones then you have only run the PSA.
3. Dell Diagnostics should open after the PSA tests, or you should see an option to open it. If you have ever reformatted your entire hard drive then Dell Diagnostics will no longer be there as an option. In that case run it from the Drivers and Utilities (Resource) CD if you have one.
4. Select 'Custom', then the audio tests. (At this point - after you have opened Dell Diagnostic custom - if you can't find the option to select the audio tests that is a symptom of a failed audio chip.)
5. The tests will run. Alternate between using headphones and listening to speakers in case only one signal path is working.
The tests will play some audio and ask you whether you heard it.
6. If you don't hear the music during the test it definitely means there is a hardware failure somewhere in the system. In this case Dell will repair the computer if it is still under warranty.
7.If you do hear the audio during the test but not when you boot back into Windows, that almost always means the hardware is okay but there is a problem with the audio driver or with Windows.
8. Error codes: If you answer 'no' to any question that the test asks you, the test generates an error code that can tell a Dell technician that you answered 'no' to that question. The test does not know the exact component that has gone bad. Before answering 'no' to a question, make note of the circumstances at that point so you will know in general what the error code means.

I have the same problem with Inspiron 9400.  My child was playing on a kid website then suddenly... the laptop speakers stopped working, although the headphone jack remained working.

I ran Dell Diagnostics and hear all the songs except in MonoOutput Sound Playback Test...I do not have a microphone on my laptop RE: above step 7.  Now, I know the hardware is okay but wonder what to do to correct the sound problem... if there is a problem with Windows? (or it could still be the audio driver, I guess)