Is the bolt broken off flush to the part/block? If not, hit it with penetrating fluid several times and let it sit overnight (periodically reapplying) and try to grab it in the morning with vice grips. have a helper hit the centre of the bolt with a punch and hammer to help shock it loose as you work it back and forth slowly.
"Easy outs" are often a mis-nomer. They are very hard steel in order to grab the comparitively soft bolt. The problem with this is that they are very brittle because they are so hard. Break one off and you'll know what "hard to drill" really is. About all that will cut an "easy out" is a real cobalt bit, and they ain't cheap or easy to find. If you use and easy out, take your time and go slow, apply lots of pentrating fluid and let it sit first. If the bolt was frozen enough to twist the head off, it will most likely break off the easy out if your don't prep it first be loosening the corrosion in some way (penetrating fluid, heat, shock, etc).
There's also options of welding to the broken bolt and drilling the entire bolt out but these may be past your abitities at home.
If you do elect to drill the bolt out, make sure you use a centre punch to start in the middle of the bolt and ensure you keep the drill square to the surface to keep from ruining the threads in the block/part.
And Indiana is right about rest. When you feel tired and frustrated, it's time to walk away. To "push on" when tired and frustrated only means more broken parts and possible spilling of blood and profanity.
He's also right about purchasig the proper tools, but I'll use any excuse to add another "shiny" to my roll around anyways!
God I LOVE buying new tools! (no stupid comments now ladies and gentlemen, tools are not to be taken lightly)
Good luck
BJ