Shotgunning is recommended by those who never learn to those who
don't want to learn. Smarter, faster, and less expensive is to buy a
tool sold even in Kmart. Don't even remove the battery (since better
information means battery still connected). Measure that battery with
the $20 multimeter. For example, if the battery is a 3.0 v lithium
coin cell, then battery is perfectly good until voltage drops below
2.6 volts. If battery is at 2.8 volts, then consider a replacement in
the next six months.
Now you have a tool to find other reasons for no power. The less
than two minute procedure is posted in "When your computer dies
without warning....." starting 6 Feb 2007 in the newsgroup
alt.windows-xp at:
http://tinyurl.com/yvf9vh
Most important in your case are numbers on purple, gray, and green
wires both before and when power switch is pressed. To obtain better
informed replies, post those measurements here. To learn further
information from those numbers, get the better informed to reply.
Remember, a power supply is only one component of the 'power supply'
system. The naive will immediately replace a power supply rather than
discover which 'system' component is defective.
And finally, using that meter will result a significant
understanding of what happens when that power button is pressed.
Those who recommend shotgunning could not be bothered to know which is
why some will recommend spending more money and more time swapping
parts until half the computer is replaced.
Get the meter. Get numbers. Get replies that will identify the
suspect the first time. Save time. Save money. Even solve a future
problem before it can create a failure. Get the meter and first learn
what has failed.