It sounds funny. When you add a web part to a wiki page and once you are done with the configuration, save the page and the web part disappears :-) Boom!!! Vanished!!
Sounds interesting and funny though, but is troublesome. The resolution is to save the page before you do the configuration part.
Once the page is saved with the web parts on it, you should be able to configure the web parts without any issue.
Happy Share Pointing :-)