Total run time each day will be about 6 hours @ 2W... Thus you need to generate more then that during daylight hours and also factor in cloudy days... You can look at a table to get the average daylight hours in your area, but across the board using a number like 4 or 5 hours a day is a good guess starting point...
So lets use 4 hours of light, you will need to generate an average of 3 Watts or so for that four hours to break even assuming no loss, but there is loss in the real world, and there is also the fact that you will not be in 100% sun for that 4 hours...
All in all you will likely need a a 5W or so solar panel to break even, likely a higher Wattage if you want reliability long term...
Your batteries should have a mAH that is factored to be able to run for a 24-48 hour period with no light (no charging) just to compensate a week of cloudy days where you will not peak out the batteries...
As for what you need I would recommend you get a commercial solar charging system that is properly rated, in many cases they come bundled with the panels...
