Those jaws are probably hardened, even if you manage to drill them its unlikely you will be able to thread them.
I would make some soft jaws if you cannot buy any specifically for that chuck and go from there.
Best to just buy a suitable chuck, but I have seen soft pieces welded to chuck jaws such as you show, and yes, jaws are quite hard, can be drilled with carbide, but not tapped; another possibility is to anneal the jaws, then they could be drilled and tapped, for your application having them left soft would not matter so far as possible wear is concerned, and yes, leaving the chuck wrench in the chuck is the biggest NO-NO in shop work, it is just not done, if the machine is accidently started the wrench goes flying and what it hits, suffers.
Please folks, I am aware of the dangers of leaving a chuck key in chuck , but this chuck has been on this lathe one time and you saw the pic of it. This was only to take a pic to post here. I apologize for posting such a stupid thing.
My rule is that my hand stays on the chuck key when it is in the chuck, whether drill chuck, lathe chuck, or whatever. I must admit to throwing a lathe chuck key once, luckily it crashed into the chip pan. Scared the hell out of me, and got worse the more I thought about it. It will NEVER happen again, and I hope the same for you and for everyone. Habits, good or bad, are repeated...
I've had some luck with local heating. Get a junk piece of steel and jam it in with a drill press until the end becomes red hot. A dull piece of carbide might also work. The last time I tried this, however, it did not work. The steel may have been difficult to anneal. You don't have to anneal it back to full soft in order to make it machinable. I have even successfully been able to locally anneal 5160, which can air quench in small sections.