Assuming the drill bit you broke was a standard HSS drill, you could use a harder grade drill bit such as Cobalt steel (or better if you can find one), with lots of cooling, to drill out the broken drill. You will also need a steady hand or a means of keeping the drill square and centre whilst drilling.
The dremel sounds a very good idea if you can get a hard enough grinding burr for it, again it needs to be harder than the broken drill material. A Diamond burr would be good but I bet they are pricey and hard to find.
Patience is key, rushing the job usually causes more trouble than taking your time. good luck with it.