Do you have a substitute Power Supply you can use for a test? May sound inane, but there are those who have an unused computer sitting around due to upgrading, or what have you, and the Power Supply is good, and compatible.
(Compatible: Is the correct size and shape to fit in the computer case, and has the minimum required power cables)
A bad Power Supply with a weak voltage rail will emulate the conditions you have described. Enough power to light LED lights, and maybe spin fans, but not enough power to turn the Processor on, or keep it running.
1) IF all the LED lights were on at once they would use less than 1 Watt of power.
2) EACH fan uses 2 to 3 Watts of power.
3) A typical Processor can use 51 to 125 Watts of power.
AMD Athlon X2 5000+?
Can use up to 89 Watts,http://en.wikipedia.org/wiki/List_of_AMD_Athlon_64_microprocessors#.22Windsor.22_.28F2_.26_F3.2C_90_nm.29
Am I stating to run out, and buy a replacement Power Supply if no substitute is available for a test unit?
Test the voltages of the Power Supply when plugged in, and computer running.
This will require a multimeter. An inexpensive multimeter can be purchased for around $8 to $12.
They are available in a multitude of stores. An auto parts store is one example.
Multimeters come in Analog, or Digital. Doesn't matter which kind you get. I prefer Analog with a needle and scale.
The Red probe lead is the Positive lead.
The Black probe lead is the Negative lead.
The Positive lead always touches the power wire. The Negative probe lead always touches a Ground (Black) wire.
The Function knob is set to DC Voltage. (Symbol is a dotted line over a solid line)
If there is a multi-position, set it to the 0-50 volt scale.
There are three main voltages to test.
1) The 3.3 Volt power rail
2) The 5 Volt power rail
3) The 12 Volt power rail.
In the cables coming from the Power Supply;
Orange insulated wires are 3.3 Volts
Red wires are 5 Volts
Yellow wires are 12 Volts.
You can test ANY of the wires, as they all end in one point for each voltage, in the Power Supply.
All Orange 3.3 Volt wires go back to one central point in the Power Supply.
This is the 3.3 Volt power rail.
All Red 5 Volt wires go back to one central point in the Power Supply.
This is the 5 Volt power rail.
All Yellow wires go back to one central point in the Power Supply.
This is the 12 Volt power rail.
ALL Black wires are Ground wires.
4-pin standard Peripheral power cable,http://www.playtool.com/pages/psuconnectors/connectors.html#peripheral
The Red wire is 5 Volts. The Yellow wire is 12 Volts. The two Black wires are Ground wires.
4-pin standard Peripheral power cable unplugged, the Positive probe lead is inserted into the Front of the connector to the Red wire. (The Back of the connector is where the wires go in)
Computer plugged in, and turned on.
The Negative probe lead is touched to ANY of the two Black wires.
You should read 5 Volts.
(Shock warning: Two D cell batteries produce 3 Volts DC.
There is no shock to be concerned with. There may be however, a small spark when the Negative probe lead is touched to a Black wire. There may not be a spark)
Same procedure for the Yellow 12 Volt wire.
To test the 3.3 Volt power rail you need an Orange wire. This is in the 24-pin ATX main power cable,http://www.playtool.com/pages/psuconnectors/connectors.html#atxmain24
The 24-pin ATX main power cable is left connected to the motherboard. A paper clip is straightened out, then inserted into the Back of the connector.
The back of the connector is where the wires go in.
The paper clip is inserted RIGHT NEXT TO the Orange wire in the socket hole. The paper clip MUST go down in far enough to touch a metal terminal.
At the end of every wire going down into that connector, is a metal terminal end.
It is approximately 3/8ths to 1/2 inch down in the socket hole.
Computer unplugged from power, the paper clip is inserted down into the Orange wire's socket hole.
Computer plugged back into power, and turned on, the Negative probe lead touches ANY Black wire.
Doesn't have to be in the 24-pin ATX main power cable connector.
(However you can use a paper clip on a Black wire in the 24-pin ATX main power cable connector, also. Just be SURE the two paper clips Do Not touch each other. I would connect to a Black wire in an unplugged 4-pin standard Peripheral power cable)
I think it may be Electrolytic Capacitors breaking down in the Power Supply. Causes a weak voltage power rail, or power rails.
Electrolytic Capacitors can also break down on the motherboard,http://www.capacitorlab.com/visible-failures/index.htm
If we go along with your last statement, which leads me to believe you may think there is a problem with the integrated graphics chipset, (VIA K8M890), then use a graphics card, and bypass the integrated graphics.
Also make sure the inside of the computer is clean.
The two main reasons for computer failure is a bad Power Supply, and a dirty computer inside.
For additional questions please post in a Comment.