To answer your question, you will need the following:
Front Der.
Rear Der.
Shifters
Battery, charging cable, two wires...one from the battery to the front der, the other from the battery to the rear der.
If you want to update the firmware on the shifters the easiest way is to buy another, much longer wire and a connector to connect the shifters...one at a time...to the front der. wire at the front der. The battery is the brain of the system. Another way is to connect the wire from the shifter to the battery directly but you have to pull the seat post and if not careful or if you don't have wires long enough you could disconnect one of the der. wires. It's far easier to use the front der. wire.
The rest of the 105 drivetrain is compatible...crankset, chain, cassette, etc.
Whether you go that route or not is your choice and I have no advice. I changed from 105 11 speed mechanical to 12 speed 105 Di2 and love it. Shifting is so fast and all you need is a light touch of the controls. You can alter the speed of the rear der. shifting speed. The front der. self trims to the gear you're in.
Increased weight is a non issue to me. My Aethos is under 16lbs with cages and pedals and I've done nothing special or spent 'crazy money' on super light weight parts.
Do your homework, think about the differences and maintenance requirements, look carefully at the similarities and differences between the bikes, not just the drivetrains.
Good luck and keep us informed...