Step one, walk away from this for a while. You're not the first person make a bad situation worse by letting your desire to fix something NOW, rather than stepping back and thinking first. There's even a name for the process - tunnel vision.
When you're fresh, (tomorrow) you might try removing the broken off stud with long nose pliers, but don't go crazy trying. In all likelihood the solution is to drill through the bolt straight across to make a hole big enough for a QR unit (with nut). Most use 5mm skewers so there's plenty of material there. Measure the width across the outside of the ears and buy the QR unit first, so you know for sure what size hole you want.
When you start drilling, be sure to keep drill square. To keep the drill from deflecting at the irregular break, you're probably best off drilling from the outside toward the middle. Or you can file the broken bolt flush through the slot, then drilling right to left using the unthreaded ear as a guide. When you're done leave the remainder of the bolt in there, but coat it heavily in grease so it doesn't rust.
BTW- sometimes discretion is the better part of valor, and there'd be no shame in realizing that you're in deeper than you can manage, and letting a decent pro fix this for you.
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