Touch won't boot -- endless 15 second cycle of logo/blank screen

Hi all. I practically used to live in these forums back in 2006-2011 but it has been years since I've stopped in. Glad to see the community is still thriving.

My SB Touch started acting up recently. When I plug it in it shows the Logitech logo for about 6 seconds, goes blank for about 8 seconds, and then repeats the logo/blank cycle until I unplug it. If I put my ear next to the SBT I can hear a faint click when the screen goes blank, suggesting to me that the screen or perhaps the entire SBT is powering off completely at that point.

I've tried doing a factory reset a few times (holding reset button on back for ~15 seconds until the screen shows "Factory Settings Restored"), but no joy there.

Any ideas? I'm thinking about replacing the power supply, which is the stock unit that I got with the SBT.

I've also thought about trying to reflash the firmware, but I've forgotten how to do it. I did it a few times during the SBT beta testing and I vaguely recall downloading a firmware image, putting it on a USB stick, inserting the USB stick in the SBT, and then powering up. But I suspect there are probably a couple other steps in there that I'm forgetting.

Thanks in advance for any tips.