Old, and hope you've gotten it fixed, but in the meantime - diagnostics...
The basic rule is "divide and conquer" - keep trying to split the problem and see what changes. The basic assumption is that only one thing is broken (if several things are broken you can end up chasing your tail for a while...) The guiding principle is start with simple stuff and move to more invasive tests as the simple stuff passes...
Symptom - some lights on, but no video, no beeps on startup, no other signs of life. This suggests deep hardware issues as it doesn't sound like you are getting to even the "POST" (Power On Self Test) stage. That machine was on and died suggests a hardware failure, possibly power supply.
General process for PC troubleshooting - think in terms of the boot sequence...
Hit switch, Power supply turns on, starts spinning fans, spinning up hard drive, and other "no-brain" bits that just depend on having power. "5v good" line asserts, applies power to motherboard / CPU. CPU does internal self test, looks for BIOS, reads BIOS and follows instructions to test rest of mobo, probably show flash screen or test progress once video section reached. If problems found, beep code, or beep for pass. Look for external devices. Look for boot device (in order defined by BIOS) and follow instructions to read boot sector, follow instructions on boot sector to load O/S and boot it up. Sounds like you aren't getting to the POST point.
1. Rule out monitor - w/o monitor plugged into PC, you should still be able to get it to show power on light, and display the onscreen setup menus if it doesn't . Also try it on a different PC if it works, definitely not a problem (doesn't sound like it in this case)
2. Strip the box on the outside - unplug everything but the monitor, turn it on - you should get some sort of video display, and probably some beep codes, even if the machine is unhappy about not having a keyboard and mouse. If nothing, repeat with no monitor... Also listen for hard drive spinup, fans, power on lights, etc... If you don't hear those it looks more like power supply, but... If you get signs of life, start putting stuff back in, starting with keyboard, problem is with the thing that makes it stop working, or you have a software issue...
3. Open up box, remove everything but video card and RAM - repeat. If you get a post display, start adding other bits back in, again testing each time. If not, remove video and RAM, if still nothing problem is Power supply or mobo. (CAUTION - PSU or mobo failures can damage other parts that will pass damage back to replacements, use extreme caution when replacing...) If you get beeps (the system WILL be unhappy not to find any RAM)
then again start adding parts one at a time.
NOTE - on newer (i.e. 386+) systems, don't try to run w/o heatsink in place on the CPU, or w/o heatsink fan plugged in for more than a few seconds. Also note that some systems look to see if the fans are working and will fail if they aren't - however this should give you a POST error.
Hope this helps,
Gooserider