No way around soldering if you are using the Tower kit. If you look at your photo of the above the TWR-PROTO is that black board near the rear of the car. That picture actually has a fair bit of custom work done.
If you want a solderless option I would suggest our newest kit, but it is based around the FRDM platform, not the Tower. It eliminates a lot of the need for wiring.
You would only need two pieces to swap out of your existing kit. Instead of the TWR-XXX you would use FRDM-KL25Z. And instead of the blue rectangular motor control board you would plug the new FRDM motor control shield into it.
I will put you in touch with your regional University Program contact to discuss further.