You can also extract uranium from seawater, and use it to fuel nuclear reactors, and use the nuclear reactors to power your desalination.
A 1GW nuclear power station consumes about 1t of U per year and produces about 8000GWh of electricity in that time. So we need about 1mg/8kWh. We got 12 times that much.
So, it sort of works out. Of course, this assumes a breeder reactor and I pretended that brine is purified uranium. If breeders are commercialized, it might be worth looking into this. Right now, it isn't.
This [1] article from 2012. It presents a Uranium extraction method from seawater that would put the extraction cost at $300/kg. Nowadays the market price is about $70/kg. However, if method were to be applied to brine instead of regular seawater, maybe the cost would go down. Hard to see it going down by a factor of 5 though.