It is impossible to do this as demonstrated on the video for two reasons:
1) This is using the PlugIn feature found on older iOS devices with the 30-pin connector which had pins for analog video out. Newer devices that have the Lightning connector do not have this.
2) The video shows him in a 2nd gen MINI which still had video hookups in the AUX port. Because Apple no longer supports the PlugIn video out feature, the 3rd gen MINIs no longer have an analog video in.
HOWEVER, Bimmertech makes a retrofit kit for BMWs with the NBT headunit
(MINI calls this Connected XL or Navigation Professional) that enables AirPlay and Miracast mirroring, which works over Wifi and does not require any jailbreaking since AirPlay is built in to iOS for screen mirroring.
Unfortunately, even though the NBT headunits are similar between BMW and MINI, no one has tried it yet. It's also not an easy retrofit and requires taking the headunit out of the dash.