Actually, though, I see a major error in my calculations. What cost US$13000/tonne in 02019 wasn't lithium; it was battery-grade lithium carbonate. The molar mass of Li₂CO₃ is 73.9 (amu); lithium itself is only 13.9 amu of that, or 18%. So that's US$69000/tonne for the lithium itself, 5.5¢ per battery, and US$1.23/kg for used batteries.
In 02015 the price was only US$6500 per tonne of lithium carbonate and thus US$35000 per tonne of lithium. Most of that was being extracted from deposits like the ones mentioned above: 0.05% lithium mixed in with 99.95% water and other things. I don't think lithium extraction technology has gotten worse since 02015 (maybe it's gotten better, but probably not worse) so I think that's a reasonable upper bound to the cost of refining lithium from such dilute deposits: US$35000 per tonne, US$35/kg.
But, as I said, the batteries represent a deposit that is 35 times more concentrated than the natural deposits currently being mined. That means that it will be much cheaper to refine lithium from batteries than from the natural deposits, although of course the chemistry of the process will be different.
The revised numbers, then, are that old batteries contain US$1.23/kg of lithium (US$1230/tonne); so, if you spend 50¢/kg (US$500/tonne) to buy the old batteries, you can spend 73¢ per kg of batteries to refine the lithium, yielding 18 g of lithium. In 02015 miners were getting paid 63¢ to refine 18 g of lithium from natural deposits, so I don't know if it'll actually cost 50¢ or 5¢ or 0.5¢ to do that same refining from batteries, but it won't be 73¢.
Except at first, because it'll be chemistry Ph.D.s refining lithium a gram at a time at a lab bench, and it will cost US$1000 per gram. But then you gradually scale up the process.