It's hard for me to say exactly what you'll need since I don't have one of these chargers. You can get a rough estimate by using a power meter like a KILL A WATT (http://www.p3international.com/products/p4400.html
) to find the power draw from the wall and use it in future calculations as you learn more about the charger circuitry.
I'm not really figuring out what voltages you actually need, I'm more so doing a sanity check on the suitability of your solar panel based off the measurements you posted. From the sell sheet for your charger, it advertises that it can charge a battery in 4-5hrs. I made an assumption of a 2000mAh battery which isn't an uncommon capacity. In an ideal world, to charge a 2000mAh battery in 4hrs requires 2000mAh/4h=500mA charge current. In reality nothing is 100% efficient so you would actually require more than this. 500mA is a lot higher than the 5.33mA from your panel so you aren't even in the same ballpark.
I can also tell from the circuitry on the back of the charger that 2 cells are charged in series which doubles the individual voltage needed to charge a NiMh cell. If you look up the charge profile for a NiMh battery you will see that the charge voltage can range between 1.4-1.6 volts depending on the charge rate. If you double the low end value, you get 2.8V which is higher than what your panel is putting out.
The voltage and current both don't have to be larger than the needs of the charger since there are DC-to-DC converters that can turn a high voltage, low current supply into a low voltage, high current supply (and vice versa) but the important detail is the original power available needs to be higher than the demands of the device you're powering. This is a lesson for another time.
Since your charger has a controller on it, it will be checking for appropriate voltages and will likely terminate the charge cycle if there isn't suitable input power. There are simpler trickle charger circuits you can look up that would be more suitable for small solar panels.