LED lights are on? Fans spin? I suggest you have a bad Power Supply.
1) eMachines are noted to have low quality Power Supply's.
Saves eMachines money so they can sell budget computers.
2) A bad Power Supply is one of the leading causes of computer failure. I have, (Had actually), a shop half full of computers with this 'ailment'.
Some caused by the owners never cleaned out the inside of their computer, nor the inside of the Power Supply, with a can of compressed air for computers on a regular basis as needed.
3 to 5 years without cleaning the inside of their computer.
Mice could have made nests inside easily, with the 'Gunk' that was there. A bald eagle could have too. Rant off.
Others, caused by the computer manufacturer used a low quality Power Supply.
Low quality components used inside.
One of the major components that fails, is caused by a particular Electrolytic Capacitor that is used.http://en.wikipedia.org/wiki/Capacitor_plague
3) Power Supply's used in PC's are a type of SMPS.
(Personal Computer. Statement holds true no matter what type of personal computer it is)
[ SMPS = Switched-Mode Power Supply
PSU = Power Supply Unit
All terms relate to a Power Supply used in a PC ]
A bad Power Supply will have a weak voltage power rail.
There are three major power rails used in the ATX form factor Power Supply, for an eMachines T6420 desktop computer.
A) The 3.3 Volt power rail (DC)
B) The 5 Volt power rail (DC)
C) The 12 Volt power rail (DC)
(Two D cell flashlight batteries produce 3 Volts DC)
A Power Supply with a weak voltage power rail, may have enough power to light LED lights, and spin fans, (or spin a few times, ans stop), but won't have enough power to turn the Processor on.
1) ALL the LED lights combined use less than 1 Watt of power
2) EACH fan uses 2 to 3 Watts
3) A typical Processor can use 51 to 125 Watts.
Depends on what Processor it is.
Uses an AMD Athlon 64 3400+, processor.
[The motherboard used is an MSI ms-7207
Also known as K8NGM2-L ]http://www.emachines.com/support/product_support.html?cat=Desktops&subcat=T%20Series&model=T6420
This Processor can use Up To 89 Watts.http://en.wikipedia.org/wiki/List_of_amd_athlon_64_processors#.22Newcastle.22_.28CG.2C_130_nm.29
No Processor running, No computer.
No computer = No Video Signal
Suggest using a KNOWN to be Good, Compatible power supply, for a test unit, if available.
Should be the ATX form factor.
Approximately 6 inches Long, (15.24cm), 6 inches Wide, (15.24cm), and 3-1/2 inches Tall. (8.89cm)
Correct amount of power cables, and the correct type.