Center punch the broken-off bolts as close to center as possible, then use a center drill to start the hole:

http://www.use-enco.com/CGI/INSRIT?PARTPG=INSRAR2&PMAKA=369-1840&PMPXNO=16770031

Follow up with increasing sizes of high speed steel screw machine drills- - - -they're short and stubby, andd less likely to break. They're also very high quality high speed steel construction:

http://www.use-enco.com/CGI/INSRIT?PARTPG=INSRAR2&PMAKA=302-2014&PMPXNO=2618939

ENCO has been in the machine shop supply business forever- - - -I've done business with them for at least 30 years. Good prices and quick service!
Jerry


"It is better to be silent and be thought a fool than to speak and eliminate all doubt!" - Abraham Lincoln
Cringe and wail in fear, Eloi- - - - -we Morlocks are on the hunt!
There is nothing noble in being superior to your fellow man; true nobility is being superior to your former self. - Ernest Hemingway
Love your enemies and drive 'em nuts!