The iRig sends MIDI either through the lightning connector or through USB. The MIDI bit does have USB MIDI in, but does not send power. In order to control the bits you would need to find a way to connect to the USB MIDI in on the MIDI bit while still getting power to the iRig.
This presents an interesting problem that Arturia figured out with the beatstep pro that includes a USB splitter that allows a wallwart USB power supply while still sending MIDI through a Y split. You may be able to use a powered hub with the MIDI bit as the host that would allow the iRig to get power from the hub and potentially sending the MIDI info along to the MIDI bit, but I have no idea whether this would work. In theory it should work, but you will need an adapter that takes the full size male USB host plug down to the micro size for the MIDI bit.
You could also use a computer as a go between with a DAW that can send the MIDI through one of the unused USB connection from the computer. That is a bummer though unless you are always going to have a computer with you. There are also specific midi I/O modules like the iRig 2 that has normal I/O/T MIDI connectors.
As a last resort you could just get a different keboard midi controller that doesn't rely on the usb connection for power or has the split cable that Arturia uses.
It is definitely possible, but you will need something to solution for the power issue with the iRig. It would be a lot easier if it didn't rely on the host device for power