You can check the online sources for "solar insolation" for your state and seasons ie winter.
At higher latitudes a solar panel might only get average 3 hours per day at full output, maybe less if you get a rainy spell, and a 5w panel might not make 5w with a weak winter sun.
I would worst case everything, ie allow for 15 hours of light on = 15x0.08A = 1.2ah
a battery good for 4 full days no sun = min 6ah
So a 5 watt panel at 3 hours effective duty is 15wh, sounds ok but doesn't quite have the ability to power the energy requirements AND recharge the battery. It might cut it provided you dont have a week of solid winter rain etc.
Since the next alternative is probably a much more expensive 10watt panel I would probably try the 5w panel and a slightly larger battery, say a 10or 12 ah battery.
You may need a charge controller since the battery will be charged during the day and the load is at night (which is always a worst case scenario). You can make a simple power zener with a 13.2v zener diode and a transistor on a heatsink if you don't want the complexity of a proper charge controller.