Helicoil comes with a special tap and a drill. You enlarge the original hole with the damaged threads using the drill included in the Helicoil kit, then use the special Helicoil tap to rethread the hole and then screw the Helicoil into the newly threaded hole. Helicoils are made of stainless steel and are tempered so they have spring tension to hold them in place. In addition you can carefully peen the circumference of the hole around the Helicoil using a center punch to prevent it from backing out next time you remove the fastener.
Re: Removal of the broken screw-
On various occasions I've use methods other than screw extractors to remove broken screws. In truth I avoid screw extractors if at all possible. One of the main problems with screw extractors (aside from the problem of drilling on center) is that 99% of them are tapered and when the enter the broken part apply outward pressure on the threads on the hole increasing friction and making the broken part more difficult to remove. There are straight flute, non tapered screw extractors and some sets even come with drill guides. They have splines which cut grooves into the broken screw which decreases outward pressure, but there is one problem common to all screw extractors; they are by necessity made of hard steel much like a tap, which also makes them brittle and easy to break and once broken difficult to drill out and remove before continuing with the original problem.
Spline Screw extractor set: (I am not recommending this particular set. It is a knockoff of an expensive USA made set and I know nothing about the quality of this set)
https://www.amazon.com/Neiko-04200A-Drive-Guides-Extractor/dp/B006YDQ812/ref=pd_day0_hl_469_4/136-4587704-9658913?_encoding=UTF8&pd_rd_i=B006YDQ812&pd_rd_r=c6949605-7ab0-11e9-ab53-6922a0711ce1&pd_rd_w=6502j&pd_rd_wg=wT04F&pf_rd_p=ad07871c-e646-4161-82c7-5ed0d4c85b07&pf_rd_r=VKQC07V8V9A2KGMFBYQA&psc=1&refRID=VKQC07V8V9A2KGMFBYQA
One advantage of a set like this, or any other straight flute type, is that works on right and left hand threads.
Method #1- Take a nut of the appropriate size and place it over the broken screw. Use a MIG welder to weld the nut to the screw through the hole in the nut. Then use the nut as if it were the head of the screw and screw the broken piece out. This method works very well becasue welding also heats the broken screw and releases any corrosion that might be holding it in. (Small screws such as this one may not have room for this method)
Method #2- Heat the area around the broken screw and use your favorite rust buster on it. You may even try heating it a couple of times adding rust juice each time. Such heating will help the juice get down into the threads. At this point I've used a hard pointed end of a scribe on the circumference of the broken screw to try to turn it out. Once the broken screw has been backed out enough you can grab it with needle nose pliers and back it out thee rest of the way.
Method #3- Similar to the above but instead of a scribe a sharp prick punch is used along with a tiny hammer to tap the broken screw in the counter clockwise direction (for right hand threads)
Method #4- I've used a Dremel tool and a tiny cut off blade to cut a groove in the broken screw and used a Keystone (slotted type) screwdriver to simply unscrew the broken screw.
Method #5- This one requires some special equipment that many people don't have, a lathe and a fractional and number drill set.
If the broken screw sits above the surrounding surface even a tiny bit, in the lathe I'll turn a piece of steel large enough to fit around the broken screw while resting on the surrounding surface and through drill the center of that piece of steel for a drill bit that I intend to use to drill out the body of the broken screw. The drill chosen should be as close as possible to the minor thread diameter (found in handbooks) of the broken screw. Then I counter bore about 1/8" deep to the outer diameter of the broken screw. The whole affair ends up being about an inch or so long. In effect I've made a drill guide which if used carefully will assure that the drill is centered on the broken screw so the surrounding metal will not be damaged. The end of the drill guide with the larger hole is placed over the protruding broken end of the screw and held there tightly by any method possible. Use the minor diameter sized drill to peck drill (preferably in a drill press or vertical mill) through the guide to the bottom end of the broken screw. What is left is a tiny ribbon of thread from the broken screw in the threads of the parent material. At this point you can pick the end of the "thread ribbon" out with a scribe, grab it with needle nose pliers and pull it out of the hole with no damage to the original threads.