You don't.

Here is what you do:
1) Buy an LCD with desirable size and case.
2) Fabricate a mount that you can slide the mini into from above and lock without blocking the optical drive or ports.(with the optical drive facing out)
3) Get a USB hub and attach that to the display as well so it is easily accessible.
4) Done.

Alternatively you have the option of replacing the LCD panel in the iMac if you find one of the correct size and aspect ratio. That would be a pita however.