Torres del Paine (Chile) is amazing, but so is the El Chalten area (Argentina). They both offer comparable world class hiking and Patagonian scenery, but if you find that the logistics of planning Torres del Paine at this relatively last minute are difficult, definitely consider El Chalten which may be easier to plan.
You basically fly into FTE (El Calafate), and then take the bus to the town of El Chalten.
From El Calafate you can also visit the Perito Moreno Glacier (most famous in Patagonia).
Flying within Chile or Argentina is cheap and easy. But I've usually found flying *between* them to be difficult and/or expensive. Their flights always route back to their respective country hubs (SCL or EZE/AEP).
In Northern Chile you also have Atacama. Pretty incredible red rock / Mars type landscapes, and some of the best stargazing in the world (Can also tour the ALMA space observatory). And salt flats. And geysers. And colourful mountains. And there's a mountain there named Cerro Toco. You *start* the hike at 5200 metres elevation. The hike itself is only 2 km long, and ascends to 5600 metres. (Don't hike this alone, altitude sickness is certainly a danger)
Northern Argentina also has beautiful landscapes.
But yeah, with 22 nights, some tough decisions will have to be made! You could easily fill 3 weeks in any one of those countries.
A map of Northern Chile & Argentina showing all the things to do/see if you happen to head up there:
https://tinyurl.com/y4jn47db
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