Suggest you replace the Power On Switch Board.http://support.dell.com/support/edocs/systems/ins6400/en/sm/index.htm
1) ALL power removed from the laptop. Remove the AC adapter (Charger) and Battery.
2) Remove the Hinge Cover. (Refer back to the Dell service manual above - Hinge Cover)
3) Remove the Keyboard
4) Remove the Display Assembly from the laptop
5) Remove the Palm Rest
6) One example of the Power On Switch Board,http://www.maksent.com/servlet/the-920/DELL-INSPIRON-6400-POWER/Detail
[ You may wish to also buy, and wear an ESD wrist strap.
(Electro Static Discharge )
Connect the alligator clip to a good ground source. I connect to an unpainted surface, of the metal frame of an open, empty desktop computer case.
Average price of an ESD wrist strap is $3 to $6.
Also use a multi-compartment container for the various screws you will remove.
Label each compartment for the area the screw/s come out of.
If no multi-compartment container is available, substitute with a clean egg carton. Label each egg holder.
SOME screws look VERY similar to other screws.
Do Not mix them up ]
For additional questions please post in a Comment.