As the others have said, what you want is readily available. A direct answer to one of your questions is, yes, you can access the internet wirelessly with the laptop without the desktop running. The others did a good job of laying out how the wiring is done.

One issue not mentioned is how the high speed modem is controlled. Your sister will have a program running on the desktop right now that is similar to the utility you would use for regular dial-up. When you go the wireless router method, you don't use that utility anymore. The router has it built in. You control the router by accessing it from a web browser on the desktop. All the login settings are entered on the router. Settings for controlling the dsl modem will be one screen, while settings for the wireless will be on another screen.

One more thing. She will be able to access her printer from the laptop, but the desktop will have to be turned on for that plus it will need some settings adjusted.

Wireless security will be an issue. Spend some time learning about it before leaving her to the wolves.