Your drawing is a little confusing. Here’s what you need:

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There’s nothing special about a setup with a light in the button. It’s literally just an incandescent bulb across the button. If you connect your circuit across the button terminals, it’ll work fine.

The 4 diodes in a loop is a “full bridge rectifier” that gives you DC from AC. The 5V regulator could be something like an LM7805.

The “large capacitor” is to keep power applied while the button is pressed, though you might be better off in that case with some small onboard battery. You just have to make sure that your battery charging circuit doesn’t draw too much power.