Suggest you start with checking the receptacle in the surge protector, that the Computer is plugged into.
I have had 3 surge protectors now that just THAT receptacle was bad.
"Hmmm, surge protector LED power on light is lit.
Monitor, printer, and router, plugged into the surge protector have power."
Plugging in a lamp into the receptacle in the surge protector, that the computer was plugged into, proved just THAT receptacle was bad.
Not an issue?
Then I suggest you check to see if the problem is the Power On switch, or the Power Supply.
The Power On switch is located inside the plastic Power On button.
This is an example of a generic ATX Power On switch,http://www.directron.com/atxswitch.html
The test is to use a jumper wire for the Soft Power On, and bypass the Power On switch.
If the Power Supply comes on, the problem is the Power On switch.
If the Power Supply does Not come on, the problem is the Power Supply.
Bad Power Supply.
Dell Support, and the Service Manual main page, for the Dimension E310,http://support.dell.com/support/edocs/systems/dim3100/en/sm/index.htm
If you click on Technical Overview, then on System Board Components, you'll be taken to this page,http://support.dell.com/support/edocs/systems/dim3100/en/sm/techov0.htm#wp1058089
Here you will see an illustration of your motherboard. This is a Top view. The view you will see when you open your computer case.
(Main menu - Removing the Computer Cover)
The number 9 points to the ATX main power connector on the motherboard.
Viewing closely you will count 24 socket holes.
12 on top, 12 on bottom.
This connector is a 24-pin ATX main power cable connector.
If you scroll down the page to the heading - Power Supply DC Connector Pin Assignments, and to the heading underneath - DC Power Connector P1, you will see the Dimension E310 has a 24-pin ATX main power cable.
This is a more in-depth view of a typical 24-pin ATX main power cable, and it's respective connector, plus a view of a typical 24-pin ATX main power cable connector, on a motherboard,
(Not a Dell Dimension E310),http://www.playtool.com/pages/psuconnectors/connectors.html#atxmain24
The photo to the far Left, shows a 24-pin ATX main power cable connector, on a typical motherboard.
There is no 24-pin ATX main power cable plugged in.
The photo in the middle, shows a 24-pin ATX main power cable.
The photo to the far Right shows the 24-pin ATX main power cable plugged into the connector.
In the photo to the far Right you will see that one of the power wires has Green insulation.
This is the Soft Power On wire.
Also abbreviated as PS_ON
You will see many wires with Black insulation.
ALL of these wires are a Ground wire.
The jumper wire is connected to the Soft Power On wire, (Green), and to ANY Ground wire. (Black)
The method is to insert one end of a jumper wire, into the socket hole with the Green wire, and the other end into a socket hole with a Ground wire.
(ANY Ground wire)
Again looking at the photo to the far Right, you will see that the power wires go down into individual socket holes.
The insulation of the power wires goes down into the socket hole pretty far.
At the end of each power wire is a metal terminal. It is a female terminal.
This is what it looks like,http://www.alliedelec.com/search/productdetail.aspx?SKU=8630463
The jumper wire must go down into the socket hole, (Right next to the existing wire already in the socket hole), far enough to touch that metal terminal pin.
it has been suggested to use a paper clip.
The paper clip is straightened out, then bent into a U shape.
The center of the U shape is wrapped with black electrical tape.
The taped area is where you hold the 'jumper wire'
You may wish to wear a glove also.
The voltage is 5 Volts DC.
Two D cell flashlight batteries produce 3 Volts DC.
There may be a spark.
Hence I advised wearing a glove. Not so much for your safety, but for your peace of mind.
One end down into the socket hole with the Green wire, (Soft Power On), and the other end down into ANY socket hole with a Black (Ground) wire.
The contact period used is no longer than 2 seconds.
If the Power Supply comes on, you have a bad Power On switch.
If the Power Supply does Not come on, you have a bad Power Supply.
The Power Supply can come on, and still be bad. Weak voltage power rail.
Enough power to light LED lights, and perhaps spin fans, but not enough to turn the Processor on.
1) ALL of 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.
Post additional questions in a Comment.