I will be using two 13W compact fluorescents.
Also, how long will it take to charge the battery (assuming I am only using one)?
Nautilus Spiral Grid Battery
Provides up to 950 MCA, and 100 minutes of reserve capacity
Amp Hour Capacity: 50
Solar panel details:
A 50 Amp Hour battery can provide 1 amp for 50 hours or 2 amps for 25 hours or X amps for 50/X hours. So how many amps does a 12 volt battery have to provide to be providing 26 watts (2*13)?
Watts = Volts * Amps.
26 watts = 12 Volts * X amps
26/12 = 2.167 amps
50/2.167 = 23 hours.
Now most CFLs are 120 volts AC and not 12 volts DC like the battery, so I assume you have an inverter to make 120V AC power from the battery. The inverter will not be 100% efficient so the lights will last a little less than the full 23 hours. But certainly all night if the battery starts fully charged.
How long to charge the battery? Most solar panels are 12 volt so use the same equation to find out how many amps the panel makes.
80W = 12V * X Amps
80/12 = 6.67 amps
A 50 Amp Hour battery takes 50 hours to charge at 1 amp, 25 hours to charge at 2 amps, and 50/X hours to charge at X amps, just like the discharge calculation above.
However, the 80 W panel only makes the full 80 watts when it faces directly at the Sun on a clear day. It can only do that if it is mounted on a motorized stand that turns it slowly during the day to face the Sun like a sunflower. If it is mounted on a roof or fixed stand, it will only make 80 watts at (probably) noon when the Sun is directly in front of it. It will make much less in the morning and evening.
Also, batteries are not completely efficient. It takes more than 50 hours to charge a 50 Amp Hour battery with a 1 Amp current. So how much will all these inefficiencies reduce your power? It is really hard to say, but I bet at least 50%. Especially the part about not turning the solar panel to face the Sun directly. That alone if a giant loss of efficiency that most people just put up with because it is clumsy and expensive to make a motorized mount to track the Sun.