Sorry for getting in on this one so late.
I was a tool & die maker for many years so I have had to deal with broken bolts, taps and tape extractors since back in the 1960's.
MY FIRST COMMENT IS NEVER NEVER TRY TO USE AN EXTRACTOR TO REMOVE A BROKEN RUSTED IN BOLT. At best they are a 50 - 50 chance of breaking them on smaller than 1/2" bolts. Rusted in bolts should always be drilled.
Yes if you have fresh metal and break a bolt using a L.H. drill bit will sometimes yield the bolt screwing itself out by the drill bit grabbing and screwing it out.
Once you have a broken tap or screw extractor you have to get creative for sure.
Like stated in one of the comments. It is probably best to use a pencil grinder or dremmel with a carbide bit to get the center of the bolt or even use a ball carbide burr and get into the bolt far enough to guide the drill bit. You should make you a drill guide on a milling machine or bench drill press. That is drill a straight hole through at least a 1" thick piece of steel to use as a drill bushing for the tap drill bit to be used. You locate the drill bushing over the hole to be drilled and clamp in place using existing holes around the area. This keeps the drill centered and straight which is very important.
I once spend probably 6 hours hand grinding an oil galley plug out of a flathead V-8 block. The plugs were tapered and the allen socket stripped out. So I drilled the center out and then took carbide burr and slowly ground out until I could see the tips of the threads in the block then picked the pieces of the plug out with a scribe.
Now that you have the broken tap I would pull the block out. As has been stated you will need to locate a shop with a Tap Buster or EDM, Electrical Discharge Machine. Not many will have one with a Z travel enough to fit a block under the ram.
The EDM is only known by that name. It has nothing to do with a laser it is and electrical current and a dielectric fluid to wash out the burned steel.
I made a tread burning attachment for one of our EDM machines while working in the shop. Sometimes we needed to go to larger bolts in hardened steel parts. If you wanted a 1/2" -13 thread you make carbon electrodes on the lathe with 13 threads per inch but small enough to go into the tap drill size hole. The attachment I made allowed you to get the electrode in the hole and stop the downward travel and then start the electrode in an orbit that expanded and burned the threads into the hardened steel.
Now back to the block.
I see you used a 4 flute tap. Never use a 4 fluted tap. They break much easier. I saw in one post they suggested to not use hardware or harbor freight taps. I totally agree with that statement. If you are drilling out a broken screw go buy American made drill bits that are High Speed or Cobalt steel. Also buy two flute plug taps that are titanium coated. A plug tap has the threads gradually going to full depth. A bottom tap has a very short lead in and then is cutting full threads and will chip easily. The titanium coating prevents the tap from sticking as easily. Also taps come in different sizes. That is designated the the GH size. Example would be a GH-3. That would be .0015" over size. Each number up is .0005" over the nominal bolt size. I would got with GH-3.
I have used a center punch to chip out a broken tap but not deep in the hole like you have. The EDM is about the only solution you will have. You do not pound it with a 2 lb. hammer you use a small jewelers hammer and peck on the tap and chip it out.
A carbide drill bit does not like an interrupted cut like the tap will be and when you break the carbide then EDM is only solution.
The EDM will cost you probably $250 and hour a Tap Buster much less. See if you can find a fellow car nut that has a shop that might help a fellow car nut out.
I do not think you will find an EDM with enough Z travel that would be huge to fit the block under.
It looks like you might have to drill the hole oversize and plug it then drill for the correct bolt size. Would need to pin the plug in or spot weld it. Probably pin since cast iron to a steel plug.
When drilling and taping use a cutting oil not WD-40 or regular oil. They are lubricants. Cutting fluids are there to increase the cutting ability not lubricate.
When using a tap you need a tap handle and you need to use a taping block to hold the tap straight. You can make the taping block using a drill press or mill. If a 5/16" tap just drill a 5/16" hole in the block to guide the tap. All tool & die makers have a taping block in their tool box. Also always use a Tap Handle. You can get that at Harbor Freight it just holds the tap. If you try to screw in with a wrench you are putting all the force on one side of the tap and will break it. Tap handle you are pushing on both sides equal and less chance of breaking tap.
When taping go half turn them back up and about ever two turns remove the tap and blow out the chips. If the tap gets locked do not try to back it out using force. Go in and out short turns until the chip breaks off and the tap will turn free again.
Patience is key never rush it. Soak the bolts for days before you go breaking them off.
When you go back together with any fasteners use a never seize on the threads. This is on any bolts outside the crankcase. Never use inside where the oil is. It is usually glass beads, copper or aluminum flakes.
On thermoset plastic molds and die cast tooling you break lots of bolts even using the never seize. I have had to deal with hundreds of broken fasteners and taps and have always got them out.
When I work on and engine every tapped hole in the whole block gets a tap ran into it to clean the threads out and every fastener gets anti seize on it. You can get at any automotive parts house this is one brand they usually have. There is also copper flake this is aluminum.
https://www.amazon.com/Permatex-80078-Anti-Seize-Lubricant-Bottle/dp/B000FW7VGE/ref=asc_df_B000FW7VGE/?tag=hyprod-20&linkCode=df0&hvadid=312181776237&hvpos=1o1&hvnetw=g&hvrand=3035487797575239685&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9051981&hvtargid=pla-492337354522&psc=1
The good news is I’ve found both a mechanic and machine shop to get me outta this mess.
