A) The professional method:
Replace the Power On switch, that is located inside the plastic Power On Button assembly.
Using an ATX power on switch, and externally.
I'll try to keep it short, but it never seems to happen..........lol!
The plastic front of your computer is the Front Panel.
The Front Panel removes.
Access to the Power On Button assembly, is achieved by removing the Front Panel, then the Center Bezel.
(I double-space sentences in the hope, that it is easier for someone to read; using their phone)
Front view of your eMachines T2220,http://www.amazon.com/eMachines-Desktop-Computer-T2200-Processor/dp/B00006SG3Y
View of the removable Center Bezel, (Which of course, has the Power On Button assembly in it),http://www.ebay.com/itm/2111-Emachines-EMA-CENTER-BEZEL-T2200-T2240-T2542-T4080-P4-2-4GHz-512MB-GRADE-A-/321046810215?pt=US_Laptop_Housings_Touchpads&hash=item4abfe18e67
Viewing the right side of the Center Bezel image, note what looks like TABS; protruding from the right side.
These Tabs are what hold the Center Bezel, to the plastic Front Panel. They are on both sides.
Looking at the front of the computer, the left Side Panel removes.
Should be two Thumbscrews to loosen.
They are loosened all the way, and should Not remove completely.
(But if the little lock clip is missing, it/they may)
The back edge of the left Side Panel, should also have a finger-pull formed in it.
The Side Panel is pulled to the back of the computer.
The Side Panel should have Tabs on the top edge, and bottom edge.
These tabs also have a hooked end.
Due to this, the Side Panel is pulled back until it stops, then moved very slightly forward, and brought straight out to remove.
Now follow Anti-Static Procedures;
Your body carries Static electricity. Static WILL fry out (Short Circuit), the delicate hardware components inside a computer.
Relieve your body of Static BEFORE reaching inside your computer.
Computer on a table, computer unplugged from power, computer case open;
TOUCH an unpainted surface, of the metal frame of the open, computer case.
This action will relieve your body of Static.
IF, you leave your computer in the middle of working on it, be SURE to Touch the metal frame again upon your return.
I BELIEVE, that the Center Bezel can be removed, without having to remove the Front Panel.
I however may be wrong.
The tabs that lock the Center Bezel to the Front Panel, may not be able to be released; until the Front Panel itself is removed.
(I know. It's beginning to look like fun(?), huh?)
IF, the Center Bezel itself can be removed, then only the wires for the;
1) Power On switch
2) Power On LED light
3) Harddrive activity LED light,
,may need to be disconnected.
However; if the Front Panel needs to be removed, it gets a little more complicated.
The Front Panel also has two USB ports, in it; plus a Headphone jack, and Microphone jack.
Doesn't take a rocket scientist to disconnect those wires from the motherboard, but does make it a little more complicated.
So.........see if you can ease the Center Bezel out. Make take using a flat tipped tool to start.
(I'm going to 'hate myself in the morning'.........
Tool perhaps like a butterknife )
Don't get Neanderthal on it, just see if you can ease the Center Bezel off, by working on one side.
If it doesn't just pop off -> Stop.
Remove the Front Panel.
You may wish to make notes, and a drawing, BEFORE disconnecting the wires from the motherboard; as to where those wires go.
However the motherboard is an ECS L7VMM, and here is how to access the motherboard manual,http://www.ecs.com.tw/ECSWebSite/Product/Product_Detail.aspx?DetailID=61&MenuID=24&LanID=0
Click on the Download tab.
(Features / Specification / Download / Certification / FAQ / etc)
Click on the blue Down Arrow, to the right of the - Select OS - box.
Click on any Ol' Windows O/S that you want.
You are just going after the manual.
In the list click on -> + Manual(1)
(OS, and O/S stands for Operating System)
Also if you post back in a Comment, and need help in reconnecting those wires; I should be able to guide you step, by step.
Front Panel, or Center Bezel off;
The Power On Button assembly should have plastic tabs on it.
They snap into slots in the Center Bezel.
Due to the age, and constant heat, they may be BRITTLE.
I would use a hair dryer set on low heat, and with constantly moving the hair dryer nozzle back and forth slowly; soften the plastic tabs.
Then see if you can remove the Power On Button assembly.
The Power On switch, should be a part of the plastic Power On Button assembly.
At this point I use the hair dryer again, and soften it's plastic tabs enough; so I can ease the Power On switch out.
This style of Power On switch is used for a LOT of computers, and it may be the same for yours,http://www.directron.com/atxswitch.html
The non-professional approach of course, is to connect an ATX power on switch like the example; and hang it out of the Power On Button's hole, after removing Power On Button assembly from Center Bezel.
If you have never worked on a computer before, the above may seem daunting. (Also because I over detail, what could be more simply put)
However it is my belief that conversing back, and forth in Comments; I should be able to help you overcome any problems you may have with this.