Led controled with collector loosed transistor

Hi there,
I have to fix lamps for microscope. Te issue is that they do not shine all they should, so I opened it to have a look and I am a litttle puzzled.
The energy supply is 5V, the lamp is 3V, and there is a potensiometer to choose the intensity of illumination.

Here after the circuit and a photo of the thing. I had never seen a transistor used with a loose collector. It is a 13005 transistor. Does someone knows what it is for, and how could I modifiate the circuit so the lamp shines brighter ?

Welcome to ETO!
Perhaps the base-emitter junction is just acting as a diode to protect the LED against the wrong polarity supply being used?
I don't see any easy way to make the LED brighter. Can you be sure that it isn't already at its safe maximum brightness?