I went through something similar when I tried to install an aftermarket tow hitch. See
here. I ended up cutting out the broken bolt with a rotozip and then removed the remaining bolt with a
magnetic tool.. I fished a new bolt through the frame using
this. I ground a groove in the bolt to hold it in place while tightening the nut.
I used this same technique again when I broke another bolt removing my gas tank. Since I do not weld, this was the best technique I could come up with.