Thank you, Nick and BigG, I have learned a lot today. I still wonder where to buy decent 8mm drills that I can put staight through the remains of a bolt at full speed, and am surprised to learn that you can punch them out when cherry red. It's a bit late for me now, with my easily bruised hands.
you buy quality drill bits from places that supply
metal workers! They will have all the sizes you need for car work(3,6,8 and 10 will do) at affordable prices, Those four should be acheivable for less than a fiver. Ask if they have 6 and 8mm left-handed bits which sometimes wind out broken bolts as they drill, which can help for blind holes.
A properly ground, good quality 60° centre punch is worth having too, you want a deep mark so the drill starts where you want it. I'm still using an Eclipse one that was secondhand(Grandad probably liberated it from work) before I was born.
Knocking out a broken bolt is much easier if it's really hot; it needs to be glowing bright red.
Mig welding can help for a bolt broken off above the surface; you weld a blob to the end, hammer a nut over it, weld the nut to the blob and loosen before the whole mess cools down. Be careful when trying this, as if it doesn't work you'll be trying to drill through very hard mig weld.