If the keyboard doesn't briefly light up the Caps Lock, and Numlock keys, when first starting the computer, and there's no video signal to the monitor, (No Signal), the computer isn't working.
Suspect the Power Supply first.
Bad Power Supply, is one of the most leading causes of computer failure.
This is brought about usually by one of two things, or a combination of both.
1) eMachines are Budget line of computers. Cheap quality parts are used, to defray the cost of making the computer.
One of the cheap hardware components is the Power Supply.
2) Computer is dirty inside, as well as the Power Supply.
The hardware components inside a computer, as well as the Power Supply, are cooled by air from fans.
Dirt, dust, etc, cause the cooling capacity of the cooling system, to drop tremendously.
Heat = Wasted Energy.
The Power Supply tries to keep up with the call for power, cannot, and components inside the Power Supply fail.
The Power Supply has a weak voltage power rail.
It may seem that the computer is receiving power, but it isn't receiving enough.
A) ALL the lights use less than 1 Watt of power
B) EACH fan uses 2 to 3 Watts of power
C) A typical Processor uses anywhere from 51 watts to 125 Watts.
Depends on what Processor it is.
The W3506 comes with an Intel Celeron D 352
(Single core processor, that operates at a maximum frequency rate of 3.20GigaHertz. 3.2GHz)
Information on the Intel Celeron D 352 processor,http://en.wikipedia.org/wiki/List_of_Intel_Celeron_microprocessors#.22Cedar_Mill-512.22_.2865_nm.29
The Intel Celeron D 352 can use up to 86 Watts, when operating at maximum capability.
This is why it may seem as though the computer is working, but in actuality it isn't.
The Processor isn't running.
No Processor operating, no computer.
No computer, No Signal to the monitor.
No Processor running, there is nothing to make the Operating System work. (Windows XP is one example of an O/S)
No Operating System working, the keyboard isn't going to 'light up'.
From here the diagnoses should be to test the Power Supply, or replace it with a KNOWN to be good, compatible power supply for a test.
[Compatible being the Power Supply has enough Wattage, the correct power cables, and enough of the correct power cables]
The Power Supply can be tested in the computer, with all the power cables disconnected.
Use a multimeter, or a power supply tester.
Example of an economical Power Supply Tester,http://www.tigerdirect.com/applications/SearchTools/item-details.asp?EdpNo=5250576&CatId=5471
This gives information on the various power cables used, in a personal computer,http://www.playtool.com/pages/psuconnectors/connectors.html
Using the multimeter, the Positive (Red) probe is connected to the power wire to be tested, then the Negative (Black) probe is touched to a Ground wire.
There are three main voltages, given out by a personal computer power supply, (All are DC volts),
1) The 3.3 Volt power rail.
Wires that have Orange insulation are 3.3 Volt wires
2) 5 Volt power rail. Red insulated wires
3) 12 Volt power rail. Yellow insulated wires
For 12 Volt power, 11 to 13 Volts is okay. Below 11 Volts it's time for a new power supply.
4) ANY Black wire is a Ground wire.
Questions? Post in a Comment.