This web page has a list of different ways and solutions to turn on the Dell Inspiron 600m laptop PC, which is known to have difficulty powering up in certain situations, and possibly due to age, who knows. First, the official dell.com portal web page for Technical Support for Inspiron 600m and the Dell Community with blogs, forums, wikis, and more.
Other tips from all over the internet:
via Dell Community Forum > Inspiron 600m does not power up, flashing LED's (intermittent problem):
"I've been able to consistently get my Inspiron 600m started by slightly twisting the base of the notebook when I press the power button." and "according to this PAGE the three lights flashing is a system board failure."
"I called Dell Tech Support, while I was holding the line, their prerecorded message mentioned this problem. Apparently, this is a known problem to Dell. The solution that they suggested is simply taking out the battery on the back, unplugging the power cord and letting them idle for about 5 minutes and putting them back."
This sounds like Dell wants to discharge capacitors on the start circuit, then recharge when the power is reapplied, hoping that the caps will last long enough with enough charge to allow the power button to signal the start. If the cap(s) are completely failed, this won't work. You might look at the mainboard and see if you can identify any leaking or burst capacitors that might be fixed. Usually this is a factory repair or new mainboard. A step that other vendors include is to hold the power button down for a minute after removing the battery and AC power.