Step 2: The circuit
The schematic is shown in the picture.
Essentially, the circuit can be divided into the charging part to the left, the light sensing part in the middle and the LED lighting part on the right.
During the day, the voltage across the solar cell is high and current flows through the diode to charge the NiMH battery. Charging at up to C/10h amps (where C is the capacity of the battery in amp-hours) is supposedly safe for continuous trickle-charge. So with 1000 mAh batteries we should be able to handle 100 mA. Our 70 mA solar cell in practice generates 50-55 mA in UK direct summer sunlight so we are safe by a factor of 2 there - pretty much ideal for fairly quick charging but keeping the battery pack in good condition.
When it gets dark, the voltage across the panel drops. This can consume significant current from the battery (so-called "dark current", which sounds like the evil side of the force to me). Hence the diode. I have used a low vF diode to reduce how much of or energy we burn getting past it. We can tap into this voltage drop to turn on the light when it gets dark. That's where the PNP transistor comes in.
By making a voltage divider between the solar panel and ground and attaching this to the base of the PNP, we sink a very small emitter-base current when the solar panel stops pulling a voltage. This allows a larger emitter-collector current to flow. The voltage divider between the solar cell and ground can control the switch-point voltage and thus the light level at which our lamp comes on.
Once our PNP turns on, a current flows to the lamp circuit on the right of the diagram (and board).
From here we have a "joule thief" circuit for the LED light. Explanation of this is rather beyond this summary but, once again, Evil Mad Scientist comes to our rescue: see here for a great Joule-thief article and here on Wikipedia for a more in-depth explanation. The overall effect is that we light a 3V white LED from a 2.4 V rechargeable battery and can continue to use the battery as its voltage drops. The capacitor is not an essential part of the circuit but it's great for efficiency. Without it I was finding 100mA being drawn from the battery! With a 1nf capacitor that drops to around 18mA but the LED is just as bright.
Finally, the switch isolates the joule-thief part so that we can continue to charge the battery but have the lamp turned off. If you turn this off then the 5-10 mA that are generated in the shade might just allow you to charge the battery in the winter to give you light about one night a week!