Both the reset and power switches are normally-open, momentary contact to operate. If you twist the reset leads together, it's the same as permanently holding the motherboard reset button pressed. You don't need the reset button in any case, so you don't have to worry about replacing the switch.
The LED on the motherboard indicates the presence of standby voltage, so the power supply is working to at least that degree. The power switch is only a signal to the motherboard to turn on the power supply and start the rest of the system operating. So if the system doesn't come on when you press the power switch, either the power supply is bad or something is preventing the motherboard from starting properly.
A quick test of the power supply is to ground the PS_ON pin in the output connector:
Unplug the connector from the motherboard and use a piece of wire or a bent paperclip to short the pin marked PS_ON to the ground pin on either side in the same row (not across to the other row, which is the standby 5-volt pin!). If the power supply is working, it will turn on and its fan will start running. This doesn't mean the supply is 100% good, but if it doesn't start with this test you certainly need to replace it.
If the supply starts this way but not when it's connected to the motherboard and you press the power switch, then you'll need to figure out why the motherboard's not starting. There may be something wrong with the board itself, or something connected to it is preventing it from starting. Removing parts and parts substitution will help you find the culprit.
As far as getting a replacement switch assembly, you might try Gateway support to find the assembly part number and see if they offer it for sale, or check eBay. Or you can try replacing the switch by itself. On some assemblies the switch is just held into the bracket by tabs, and companies like Digi-Key
sell suitable replacements.