When you are drilling out a stainless steel screw, heat is your worst enemy. Heat will build up quickly in the area where the drill bit contacts the stainless steel screw. The heat, if not left in check, will quickly erode the tip of the drill bit. You need to control the speed of the drill bit and use a lot of pressure to keep heat build-up in check. Even while using speed control to reduce heat, drilling out the broken stainless steel fastener will take some time.
Put on your safety glasses and leather gloves.
Align the center punch with the center of the broken stainless steel screw and strike the backside of the center punch to make a depression on the stainless steel screw to hold the tip of the left-handed cobalt drill bit.
Place a left-handed cobalt drill bit that is smaller than the broken stainless steel screw into the chuck of the drill motor and coat the drill bit liberally with cutting and tapping fluid.
Align the tip of the cobalt drill bit with the depression that you placed on the stainless steel screw in step two, then flip the switch of the drill motor to reverse.
Pump the trigger of the drill motor while applying pressure on the stainless screw to drill into the stainless steel screw. If you notice smoke emitting from the tip of the drill bit, apply additional cutting and tapping fluid to the drill bit to keep the drill bit and stainless steel screw cool.
Exchange the drill bit for one that is slightly larger in diameter and repeat step five to continue drilling through the stainless steel screw. As you drill, the left-handed drill bit combined with the heat could cause the stainless steel screw to release and back out of the threaded hole. Continue to increase the drill bit size until the screw comes out.