The advantage over the original circuit is that it has constant brightness even when the battery voltage drops as it discharges, down up to the very end when its voltage goes below 2.7 volts.
The torch had seven LEDs in an aluminized plastic reflector, and a three position switch lighting zero, three or seven LEDs on successive positions. The LEDs were all connected to the 4 volt sealed lead acid rechargeable battery with independent resistors.
When the battery died, I replaced it with a 3.6 volt nicad pack from a defunct mobile phone I had lying around, and it worked, but the brightness had dropped. Replacing all resistors to raise the current through the LEDs (all seven of them) was thought to be too much of a chore, so I went ahead and wired them all in series. Now I had seven LEDs in series, requiring about twentyfour volts to light them, at around twenty milliamps.
Then I used an integrated circuit driver to raise my battery voltage to the value required.
Step 1: The schematic
TI make the circuit, so they are the people to tell me how to use their chip. It looks simple: Two capacitors to store energy and decouple the supply voltage and output voltage, a rectifier, an inductor, and finally a resistor to set the LED current.
According to the datasheet, as long as the other components are correctly chosen and wired up right, the LED current depends only upon the value of the resistor Rs in the figure below. It doesn't matter how many LEDs there are, and what voltage the battery has.
This particular integrated circuit can drive LEDs in series up to 28 volts, and the input voltage has to be in the range 2.7 volts to 6 volts. Since a white LED requires around 3.5 volts, upto eight of them in series may be used. Since the minimum voltage has to be 2.7 volts, at least two NiCd cells in series would be required to drive it - or one Li-ion.