As AC pointed out below, this cost likely includes the design, build, launch and maintenance for the satellite. Before Space-x The launch alone could have been a tenth or more of that total $13B, as most weather satellites are around 3000 kg (http://noaasis.noaa.gov/NOAASIS/ml/genlsatl.html), but with Space-X's projected costs per payload ($850/lb from Delta Heavy's $8600/lb) (http://www.nss.org/articles/falconheavy.html) this cost likely can now be in the single $M range.
While economies of scale would likely get those drones into the range of cost you suggested, it certainly wouldn't take into account the cost to maintain and monitor such a system. The congressional research service (CRS) (http://www.fas.org/sgp/crs/homesec/RS21698.pdf) identified that for operation (facilities, maintenance) it can be at least 100% or more of the cost of the drone, So that would have to drop the number of drones available to 140,000. Secondly, all drones, by FAA mandate, are required to be a operated by a licensed pilot. I would imagine the training and licensing involved for this would not be cheap, as last estimated the number of pilots was ~598K in 2009, with only ~320K certified with instrument ratings http://en.wikipedia.org/wiki/Pilot_certification_in_the_United_States), and It's likely commercial air pilots would have to have a pretty big incentive to go (http://www.payscale.com/research/US/Job=Commercial_Pilot/Salary) but keeping it on the low scale, that would have to be $50K per pilot per drone, making even a yearly cost of operation at $7B (140,000 drones * $50K/pilot). That doesn't go into operation times either, as drones are listed to operate from 10-48 hrs (CRS reference). So turn-around times for getting those drones back up would end up having even less drones available at any time for weather surveillance.
However, looking at a combination of mini-satellites might be the best option, as redundancy and low cost could take this project down by a large amount (~300K per satellite) (http://www.hawaii.edu/offices/op/innovation/taylor.pdf) . While it might end up with similar issues as stated above, there would be significantly less satellites needed based on the larger surface area covered from their height (50 km for possible best drone (http://blogs.scientificamerican.com/guest-blog/2012/02/24/high-altitude-surveillance-drones-coming-to-a-sky-near-you/)) and 870 km for satellite (http://noaasis.noaa.gov/NOAASIS/ml/genlsatl.html). But this might not be available just yet for our weather measurement needs.
In Summary, it may seem like a huge amount of money, but you need to consider all aspects of the project, not just the non-recurring costs.