You can do this by editing the workflow. Find the transition that takes the issue from the "in progress" column to the "done" column and add a "transition screen" to it. Put the fields you want to ask the user about on that screen.
If I am understanding the issue correctly it sounds like you would like to use a resolution at the time the status is changed to done. By default JIRA provides resolutions for issues that can be applied when the issue is resolved or closed or done. All you need to do is update your list of resolutions to contain the resolution types you would like (Resolved, won't do, etc.) You can learn abou this here: Defining Resolution Values.
Once you do that you will need to adjust the screen on the workflow transition to contain this field. Then when the issue is transitioned it will prompt you and users can set the resolution. You can also make is a required filed if you like. Here is some information on configuring Screens and Workflows.
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