Don't cut it too close, just in case. Leave a little stump and then cut it back later. (later, as in a couple of months) If you wire the side branch up, you get a straight trunk, which is what we are trying not to get, usually.
Cut the dead stuff off, leaving a little stump, and then let the tree grow for a while. Let it get healthy and strong. Let it pop new buds elsewhere on the trunk. See what happens.
Joanie