My suggestion is that you go to one of the BoS bunkers and temporarely recruit an npc (Stitch, for instance) with decent doctor skills. Heal the broken arm (use one of those doctor bags or what do you call them) and ditch Stitch again.
If for some reason you can't or don't want to do it this way, you could just keep trying, I suppose. Just pick the character with the highest doctor skills, give that character the doctor bag, (maybe even save your game if you don't want to waste lotsa doctor bags) and keep trying.
Naturally, you could always load a previous saved game in which you didn't have the broken arm yet.
I hope this was useful.
