Actually what you want to do is fine.
You simply need to setup the Mini with the ID that you now want it to have, and then all the Apps from that ID become available to it to be downloaded for free.
Now, if you downloaded Apps with the ID currently on the Mini then those will still be available to it, but will require that other ID to update.
If you haven't downloaded anything with the Mini's Apple ID, your best bet, is to simply Wipe the Mini and start over. When you set it up again, choose the option to use an existing Apple Id, and input the data for the Apple ID you now wish to use.
Basically changing the Apple ID used on a device is fine. Transferring content between different Apple IDs is not permitted nor possible at this time.
To change the Apple ID of the Mini you can go to Settings->iTunes & App Stores -> Apple ID ->Sign Out.
Then Sign in with the other one. If you decide to download Apps onto the Mini with the Apple Id from the iPhone 5 , you will potentially lock the Apple Id that's currently on the Mini for 90 days.
To Wipe the mini and start over:
Settings->General->Reset->Erase all Content & Settings.