I hope the USB connection to the IDE case will have sufficient power to run the drive inside.
It is not advised to attempt to run an external case without its own power supply with the drive inserted off the USB bus. The current demand to drive both the electronics in the enclosure and the drive itself on the bus would be higher than USB standard specifies, so if the ports don't simply fail, data transfer will be somewhat unpredictable and may suffer stalling. Some external drives are specifically designed for ultra-low power use and may work, but a standard enclosure and drive is far less likely to.
Many systems are engineered to provide more current on the USB bus than standards require, and thus this arrangement may work, but the Mac mini is close to the standard rather than above it.