I believe weeping cherry is grafted at the top of the straight trunk, so be sure that the new growth is from above the graft and not from the root stock -- they should "weep"
Once you are sure nothing is going to grow back from them, start testing to see if the dead looking branches are really dead. They found hollow when you flick them and easily snap with dry break. Be careful because there will be some dead looking branches that actually rant dead and will have live side branches further down.
I go around snapping off the dead twiggy stuff up to healthy branches, then clean up with clean cuts with pruners. I don't like the "bob cut" style a lot of people adopt, so leave mine in a more natural sweeping manner and just keep the branch tips from touching the ground, but I guess "the bob" is more popular.
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