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When I come across a broken stud I will center punch the broken stud and drill a shallow hole (about 1/8" deep) into the stud with a 5/32" bit. Using a short burst with a mig welder (don't have tig) I fill the hole I just drilled with weld. I let allow the weld cool just long enough for the bright red glow to fade a bit and then stack another burst weld just off center of it and again allow the color of the weld to fade a bit. I work my way around the top of the stud repeating this stacking of welds until the weld is built up above the level of the block a quarter inch or so. This allows the heat to better penetrate the broken off stud.
The block stays cool and any weld that strays onto it generally will not stick or at least makes a very weak bond with it.
Then I place a flat washer onto the stack of welds and weld it in place followed by a nut stacked on top of the washer. Fill the center of the nut with weld and allow it to cool almost to room temp. Grab it with a vice grip and try to work it off then. If your weld breaks loose from the broken stud then just give it another go. Sometimes it takes quite a few times before it finally breaks loose and comes out.
Since you have a bit broken off in yours you wouldn't be able to drill the hole first but it should work with the broken piece of bit in place.
I strip the cadmium plating off of the washer and nut before welding with a 50/50 mix of muratic acid and water. The cadmium plating makes for a messy weld.