It will take some time to heal, but you will. You have a few options here. You can 'suffer' through it using your deep breathing exercises that you mentioned. You can hand express or pump (though pumping might irritate it just as much as nursing would) while he is eating from the other side and then feed him the expressed milk from a bottle or supplemental nursing system (so that he receives as much breastmilk as he normally would but only at the one breast). You can try using a nipple shield so that your nipple isn't coming in direct contact with his mouth (though this might not work either). You can also talk with a lactation consultant or La Leche League leader. I'm sure there are lots of tips they could provide you with since I'm pretty sure this is a fairly common thing to have happen.
If this happens again in the future, I have a suggestion for you. It may sound mean or something, but it isn't. When my son bites down, I pinch his nose. It forces him to let go in order to take a breath. It doesn't bother him at all. In fact, he gets a really funny look on his face like "Hey, what was that all about, mama?"
Oh, one last thing I thought of is to maybe use some lanolin cream where he broke your skin. Something like Lansinoh (spelling) would work. But, if it's a bruise, it could take several days to heal completely.