The first step is to open up the light and take a look inside. This turned out to be quite straightforward as I just needed to remove four screws and pull the top off.
The internals are very simple with color coded wires. The only surprise is that there was a round female connector fitted to a round hole on the rim of the light. This looks like it would allow the light to be mains powered but that is no good to me as there are no sockets in the cupboard I'm using the light in.
With the cover off I took the chance to measure the current used when the light is switched on. The result: 500mA! No wonder the batteries lasted for such a short time! However, I'm not entirely surprised by this result as the light was using an incandescent bulb.
As an aside - I always find measuring current really awkward as you have to insert the multimeter in series in the circuit which is tricky when the probes are just pointy metal tips rather than clips which can hold a wire.
The next step is to make this light better. I decided the easiest way to do this would be to replace the incandescent bulb with an LED. I took a look on maplin and picked out a bright white LED which runs at 3.6V and only draws 100mA to produce 18000mcd of light. According to this site 18000mcd is 2.1 lumens which compares to 850 lumens for a 60w incandescent light bulb.
The 3.6V requirement means that I only need three of my 1.2V rechargeable batteries and won't need to add a resistor. The first step was to insert a wire instead of the fourth battery.
The next step was to remove the incandescent bulb along with its screw socket (which I have saved in case I ever need them in another project) and solder an LED in its place. The only tricky part of this was that the wires in the light were stranded instead of solid core which made the soldering more fiddly.
That's all there was to it. I put the case back together, turned on the light and took a picture.
The light is brighter than this picture suggests (my phone overcompensated with the exposure) but it is a bit less bright than I had been hoping. In future I might add an extra LED or two which would double or triple the light output with a corresponding reduction in battery life (more LEDs = more current but the same voltage).
Once everything was assembled I got out my multimeter and measured the current and was pleasantly surprised to find the current was only 70mA rather than the expected 100mA. Since I am using 1700mA batteries this means I expect a battery life of about 24 hours compared to the 3 hours which I would have got with the incandescent bulb.