Originally Posted by utopia

I dunno how much experience you have of such work but its a bitch of a job with pitfalls waiting to ensnare you at every turn.
Most important is probably to get the drilling absolutely dead central, which is far more difficult than it sounds because the threads of the old bit of stud deceive your eye.
Also to drill perfectly perpendicular.
The initial centrepunch dot is extremely important to get as near to perfectly central as possible.
Then following that up with either a centre-drill (preferably) or a small plain drill is essential before getting out the bigger drills.
When you do employ a bigger drill it might be worth considering buying a suitably sized lefthand drill as this will tend to unscrew the broken stud as it cuts.
Before doing any of this, I would give the stud a good few sharp blows with a hammer and punch in an attempt to jar loose any corrosion or binding in the threads .. followed by a good soak in decent penetrating oil.
Also worth considering is a lefthand tap and matching bolt to use as an extraction device.
And also the application of heat.

Obviously enough, a drill will tend to favour wandering off and cutting into the softer ally of the case rather than the hard steel of the stud.
If the stud is stainless (or even if its not) a slow drilling speed, plenty of axial pressure and copious cutting oil will produce the best drilling performance.

Sorry if any of that is granny egg-sucking info but better to mention than not, I reckon.

And welcome along.