Outside of the Wyndham Hotel there really isn't a good option near the airport.
Last time I was in your situation, I stayed at the Royal Inca Hotel for about $30.
Not great, but sure beats sleeping in the Lima terminal. They'll arrange for a free taxi to meet you once you exit Immigrations processing, and drive you to the hotel.
You'll also get a breakfast in the morning.
https://www.tripadvisor.com/Hotel_Re...ma_Region.html