Don't know Wi-Fi from a hole in the ground, don't want to know but, 5 vdc X 2 amps X 24 hours / 1000 = 0.24 kWh's, or more simply put 2 amps X 24 hours = 48 amp hours at 5 vdc or, if you want to know what that is at 12 vdc then - 5 vdc X 2 amps = 10 watts. 10 watts / 12 = 0.84 amps X 24 hours = 20 amp hours at 12 vdc nominal.
By "wall wart" I assume you are referring to a plug in rectifying transformer?
If you are wanting to it all relatively simple and "off the shelf", get yourself two 6 volt, 220 amp hour "golf cart" batteries. Wire them in series for 12 volts nominal. Then; one PV module say 100 watt, a matching charge controller and, a dc to dc voltage converter, and most importantly - sunshine.
Lets see if I can help out a little more...
2 - Trojan T-105's - $200.
1 - Kyocera KC 130TM - $585.
1 - Specialty Concepts ASC-12/12 - $46.
1 - Power Stream PST-DAA1205(-2) - $31.
(
http://www.powerstream.com/dc2.htm )
Mis.; wire, fuses, PV mount, etc., etc. <$200.
All of this would sum up to $1,000. but, you seem like a bright lad, you should be able to take this information, do a little shopping around and, work up a system to your own liking. Just remember that RE is a sellers market, no different than any other market. Happy hunting!