It seems to me that the push butto is a On-Off switch to power the system.
Check the broken switch.
If it makes a "click", then it is On-Off.
No "click" would indicate it is a contact switch; but that will imply some kind of relay is used, which is improbable for simple circuits.
This one is rated for 10 A and is available from radio Shack:
http://www.radioshack.com/product/in...ductId=2062519