how did the button break as in it doesn't work or the button is physically broken. you should be able to tape wire to the contacts, follow the button wire. Than then 'jump' them to start the computer. If it works you can solder the wires to the contacts and a normally open Single Pole switch. I can explain in more detail if you want me to.
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You can enable, "Wake-On-LAN"http://www.manageengine.com/products/oputils/wake-on-lan.html