It seems that you're using an outdated browser. Some things may not work as they should (or don't work at all).
We suggest you upgrade newer and better browser like: Chrome, Firefox, Internet Explorer or Opera

I've tried playing in local multiplayer, but while it was OK for the person playing on keyboard + mouse, I was having a world of trouble while trying to aim + move with the controller. Plus there's no aiming cross-hair for the player using the controller which makes things even worse.

Is there any way to play this in LAN multiplayer, so we can both use keyboard + mouse?
This question / problem has been solved by Corybanderimage
I don't believe LAN is supported, just internet play. You could create a VPN, both log into it, and then play from there in your own little corner of the internet, but that's a somewhat involved process and will require a little research.