Neat trick, but I'd never get DW to stay in a "used" room.
I decided that given the flight is long and the altitude high in Quito, that we should just rest as much as possible. The side trip suggestions were great, but I'll leave those for another time.
I went to Priceline and booked a room for two nights at the Doubletree Hotel for $40 a night + tax, so this is about the same cost as a one day pass X 2 to an airport lounge.
This way we'll only be 1 mile off the airport premises, but will have a quiet place to rest and store our luggage. We saved about $500 on airfare by taking this long layover, so I still feel like I'm money ahead.