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 :)
(A picture of the broken piece would help.)
Option 1. (By far the best answer) Buy a new hacksaw. You won't be able to fix this without removing the threaded piece that still remains in the hole, then fabricating a new piece to hold the blade. This will involve adding threads to the new piece. If you don't have the tools to do this work, then you will need to obtain them, costing you more than a new hacksaw frame.
Option 2. Remove the threaded part that remains inside the hole. Usually this involves drilling a small hole in it, then you must buy a tool for removing screws that have been broken off in a hole. (Its usually called an ez-out, or something like that. A hardware store will have them.) You will insert that tool in the hole you have drilled. It will jam in place with a coarse reverse thread with sharp edges, and allow you to back out the threaded piece. It may help to drip some wd-40 in there as you are trying to back out the threaded piece.
Drilling that hole may itself be problematic, since there is a reasonable chance that part of the hacksaw was hard steel - by design to withstand the loads it must take to hold the blade. Worse, you must drill a hole into a surface that is jagged.
You may also be able to get the threaded piece out by other means, but you must do all of this without damaging the female threads in there, otherwise you will NEVER be able to fix the new piece in there.
As I said, unless you have a few important tools, this is all something that may be more easily repaired by simply buying a new hacksaw frame. Money solves everything, although a hacksaw frame is not that expensive. And if you don't know how to get that screw fragment out, then you probably don't have those tools, nor will you use them often enough in the future to merit buying them. At the same time, it is good to be able to repair things like this, so if you have the interest and the will to do this sort of work enough in the future, then go for it. But don't be at all surprised if you find it takes you considerably more time and expense than it is really worth to simply buy that new hacksaw frame.
Best Answer
cheap grinders are cheap, get one.
Before grinding erect a shield to prevent sparks from hitting the windows or any other delicate surface. (they will stick) cardboard or an old blanket would be strong enough.