You may be able to drill out the broken off tip of the centre drill by modifying what is left of the broken centre drill body. Take a Dremel or similar grinding tool with a small, thin cutting disc installed. On the end of the centredrill with broken off tit, cut a slot up the centre of it, about the same width but slightly larger than the diamter of the broken off piece and about the same depth plus a tad more.
If you now drill into the original centre drill hole where the broken off tip is embedded with this tool, it will drill down around the outside of the broken off piece to a depth where the broken off piece of HSS will be left standing proud and can easily be removed with a pliers, scriber etc.
In future, instead of buying hard to find spot drills, you can use a larger centre drill but do not drill all the way down with it in the conventional manner. Just use the tip piece to make a small indent, enough to get your small diamter drill bit started. If you use a large enough centre drill the tip piece diameter will be larger than your small drill bit and so provide a good start, with suitable care taken.