Now, after my computing situation has stabilized, I had a chance to open up the VAIO to see what could be done. There were some helpful disassembly hints on the Internet, and they were essential to avoid breaking things. The culprit almost leaped out at me. It's the tiny inverter card (above) that makes high voltage to run the fluorescent LCD back light. The Internet (again) seems to be saying this is a frequent source of problems. The HV rectifier was charred and broken, so obviously this was the problem. It's a $50 part, and I hope replacing it will solve the problem.
I suppose there's a moral here. I could have saved money, but incurred opportunity cost and risk, if I simply repaired the VAIO. I'm happy to have a new, more powerful laptop. Still, the VAIO is handy because it has some critical features (like Firewire) that make it very useful for some of my work.
Update (11/12/2010): Thanks to lcdparts.net, I have a new inverter, and the VAIO is back in operation!