Release the steering wheel adjuster and pull it as far out and down as it'll go, lock adjuster.
Remove top and bottom cowls
Remove key reader ring (unplug single wire)
Put key in (don't turn)
Use a small allen key and push down through the hole in the top of the barrel casing and turn key until you feel it push in further, keep pressure on it and pull on the key. Barrel will pop out.
You will have to remove the switch by undoing the small grub screw(s) and get the broken piece out of that carefully
Don't turn the switch, if you do make sure it's in the same position as it was before removal or the barrel will not line up when trying to reinstall a new one.
Installation in reverse order except you don't need the allen key for this bit, with the key in and correct position you will be able to depress the lock tack fully.
If the steering lock enables, get a small screw driver and push it back down till it unlocks before trying to for the new barrel in.
It's advisable to buy a brand new genuine barrel ($40-$60 on ebay) and recode it to your key, using a second hand one risks having the exact same problem as they wear and it's a weak point that will break again.