The lower connector - the small one - looks like a 4-pin connector for lightweight utility-type trailers. If so, it will contain stop lights, directionals, and running lights, plus ground. These four are needed for the TM, so there may be an adapter that brings these four into the 7-pin Bargman. But you still need 3 more. These 3 are the wire to the backup light, the 12VDC power wire to charge the TM battery, and the electric brake wire coming from the brake controller to the TM's electric brakes. It is likely that all three will need to be new wires that must be strung from the engine compartment area to the Bargman connector. The folks at U-Haul should know how to do this. It is a common everyday task for them.
The upper connector - the large one - looks like it is not intended for a trailer. My guess - and it is only that - is that it is the Kia's connections from the main wiring harness in the vehicle to the various individual lights in the rear part of the vehicle. I don't think that any of the three new wires described above are part of this connector, although the backup light wire might be in there somewhere. Please let me know if I am wrong - it is a useful piece of information.
The task you are facing here is annoying and no doubt expensive, but it is in no sense unusual, unique, difficult, or TM-specific. The diagram in my link above is the standard hookup for a 7-pin Bargman trailer connector - they are all hooked up that way. You will be pleased to have it done.
Bill
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