Little AA battery cells are way too small to power 34 LEDs for more than a few hours. If the LEDs each have 15mA then the voltage of three AA alkaline cells in series will quickly drop to 3.75V which is the same voltage as three Ni-MH rechargable cells, and 2500mA/hr cells will last for about the same amount of time, about only 5 hours.
You could make the LEDs dimmer, then three AA cells would power them all night.
The solar panel will need to be huge and expensive to fully charge a 2500mA/hr battery.