Low fuel indicator light

[NicoB16A2]

Hi. My low fuel indicator light on my 1998 Civic is no longer coming on when it should. I tested the light by shorting the connector from the fuel-sender unit (as indicated on the service manual) and the light is ok, so the problem is in the fuel sender unit.
I removed the fuel sender unit, hoping that I would be able to notice something and fix it. Turned out I didn't

In the attached picture the fuel sender unit is shown. The blackbox on the lower left is responsible for the fuel gauge, and there is something (that i circled in yellow) that looks like a small battery connected to a yellow wire which is responsible for the low fuel light. That's as far as I got.

The question is: Does anybody know how the sensor for the low fuel light works, or what is it exactly? Any way that i can make it work again?
Can i replace that alone with any standard electrical part?

Thanks to anyone who can help me. I'm trying not to replace the whole fuel sender unit because it is (to me) way to expensive.

Nicolas
(from Argentina - excuse my bad english!)

[NicoB16A2]