The fact it broke off means the screw is probably very stuck. I would first use a penetrating oil (though WD-40 might also work), and let it sit for awhile, before trying anything further.
Hindsight is 20/20 of course, but it would have been good to do this first. If you see any corrosion or rust, or suspect it's needed and have time, this never hurts to do.
You may be able to use a small drill bit to extract the broken extraction bit, and then try again -- but be careful as you may damage the screw beyond my next tip..
If you can't get the tip out (or don't want to risk trying), I would probably use a dremel cutting wheel, and basically turn the screw into a slotted screw. This can be difficult -- not deep enough, and you'll just strip the screw worse trying to get it out, and too deep and you'll weaken the head so much it'll probably snap off when you try to turn it.
The next thing beyond this is to use a "reverse drill bit" or "left-handed bit". You can drill right into the middle of the screw itself (with your drill in reverse), and the bit should actually come out before you've gotten all the way into it. Use a bit that is smaller than the thread size, otherwise you'll strip the threads so you won't be able to put another screw back in. Again, judgment is necessary here, too small and it won't get enough grip, too big and you'll strip the threads -- though maybe by this point you don't care that much :)
![alt text](https://i.stack.imgur.com/asVqf.jpg)
The bigger question is how many vertical supports and how long are they? 1/4" lags 2” into the 2x4 has 510lb pull out strength (255lb/inch) put 2 or 3 in each piece of strut and each one can handle hundreds of pounds.
5/16” lags 266lb/inch; 3/8"lags 305lbs/inch. You don’t need a very big lag bolt when using multiples. My examples have a small safety factor added in and these won’t degrade the strength of the 2x4. I usually drill pilot holes,
But not all brands of screws need pilot holes. The depth into the wood is the threaded part that provides the pull out strength over ½" sheetrock I would make sure to have at least ¾" -1” smooth shank that provides the sheer strength as the weak point is the start of the threads.
Best Answer
Knock out the drywall around it until you can grab it with locking pliers. Either unscrew it or break it off (if it broke going in it may not want to come out any more cooperatively.) Spackle the hole and move whatever it is that caused you to drill it up or down a little bit so the new hole does not line up with the old one if the screw is broken off in place.