If you don't mind using a Mac, buy a second hand
IBook. They can be had for very little money and don't break easily. On the downside there might be compatibility problems with your virtual desktop software, you've got to learn a whole new system to use, and they look like they're made of duplo.
If you want to stay with windows I'd probably go for a second hand Toshiba laptop. They're simple and reliable enough. Get a techy friend to install windows 2000 over the top of the existing operating system.
Don't worry about your keyboard problems - it's easy to get an adapter that will let you use a traditional keyboard in a USB port.