Do I get scrabble points for the summary?
Start with a liquid dessicant (such as CaCl dissolved in water, like my earlier idea), run it through a device to allow it to absorb water from the room air. Nothing strange so far :)
Run the now dilute liquid desiccant through a high efficiency countercurrent
heat exchanger, to heat it up.
Pass the liquid desiccant through a hydraulic motor, moving the liquid from atmospheric pressure to partial vacuum, and extracting mechanical energy from the pressure difference. (You could use an expansion valve instead, but that would be wasteful of energy).
Due to the low pressure, the some of the water in the warm dessicant boils. Due to the heat of vaporization, the remaining liquid has a lower temperature -- probably close to what it was before it was warmed by the heat exchanger.
At that lower pressure, pass the mix of steam and concentrated liquid desiccant through a gas/liquid seperator.
Move the liquid out of the gas/liquid seperator using a pump, and send it back to the component which exposes the liquid desiccant to the room's air.
Move the steam out of the gas/liquid seperator using a compressor (moving it from partial vacuum to atmospheric pressure), and send it back through the high efficiency, countercurrent heat exchanger, and then either into a collection container, or down a drain.