Size of lock nut of front hub? Where to get it?

I broke a lug on the front wheel when I tried to loosen the lug nut.
To replace the lug/hub, I started disassemble the front hub. Now I am
stuck by the lock nut of the front hub. I tried the largest socket
(27mm) and it still won't fit.

What's the size of the lock nut?
Where to get a new lock nut?
After I take off the old hub, should I change the hub or ask a garage
to replace the broken lug? Which is more expensive?

Appreciate your advice

Tingfang