On a Wiki page, the web part always vanishes after the configuration is done.
This is because when you edit the web part, you go to another page without first saving your wiki page. When you come back to the page after configuring the web part, the last saved version of the page is displayed. It doesn't have the web part on it.
When you add a web part to a wiki page, save the page BEFORE configuring the web part. If you need to add several web parts, you can add them all and then save. Once the page is saved with the web parts on it, you should be able to configure the web parts without any issue.