As a mainboard replacement costs almost $100 U.S., short of warranty service (if you qualify) your best option is probably to look for someone selling a bricked Archos 32 like this one
if you're bold with the DIY approach to repair. And yes in 'Settings' > 'About' option at the bottom, there is a 'Power management' option that lets you turn on 'Deep sleep' that does a decent job of conserving the battery and may be your best bet unless you want to be among the first to document Archos 32 internals.
If I were in this situation I'd probably hold out and use power management to get buy until I could find a web page documenting how to properly open an Archos 32 -- it's a fair bet that the switch is a surface mount affair that can't be repaired and you need a whole (and working) board replacement to get it working -- or it may just be dirty and stuck and a shot of contact cleaner might get things going again. Best of luck, let us know how you make out if you attempt surgery on your 32, documenting would be fun and educational to follow as well.