Here's a Bing search for cheap laptops under $100. I don't know anything about these, but it's a start. (A very quick glance says that their screens are smaller than 20", but again, it's just a starting point for research).
http://www.bing.com/search?q=Cheap+L...+100&FORM=R5FD

You wouldn't really need an additional keyboard for them. Wireless mice are OK I suppose, but A) you have to worry about changing the batteries (not "frequently," but still "eventually")(and I'm a cheap bastard about them! lol), and B) they cost more than a wired mouse (which can be dirt cheap for the most basic ones).