Dump the puncture repair kit and take a spare tube for commuting (you don’t wanna be patching stuff up in the rain imho). Patch the old one when you get in.
Overshoes and a decent jacket for the rain (though with mudguards you might be okay on the overshoes front, and yes, they can be used with normal shoes).
Obviously you’ll need 2 decent locks for the bike – this LFGSS thread lists decent locks which work, but they aren’t cheap: https://www.lfgss.com/conversations/144109/
Also – have you thought about how you’ll carry stuff on your bike? A decent waterproof backpack perhaps?
Once you’ve done the confidence course I wouldn’t worry about “getting into the position” of cycling to and from work – I managed it without any practice or prep and my commute is also 12km – you’ll get faster etc over time. 12km isn’t that far-- I got mine down from 50 minutes to 30 minutes in about 6 months. I’d also recommend doing the commute for the first time on the weekend as the roads will be quieter and you can work out a route. The confidence people should also be able to help with this.