I hope you get some work. Need that money to pay essentials, then a few toys. Are you skilled in woodworking, kool if you are. Good money in that. Also consider cabinet making.
Back to the broken tap. Consider reverse left hand drills bits. They will provide some torque to help remove the tap as you are drilling. Yes, you instinct is correct. Start with making a small flat spot on the break as close to the center as you can. Then take a sharp punch to make a small indentation to hold the drill bit from wondering. Then us a small bit to drill a hole and work you way up. Use that cutting oil to keep your bit sharp. If you use easy-outs, be careful as that have a habit of snapping also. Good luck.