Answer: What you describe is rather unusual growth for a tree, almost like it's in suspended animation. Pruning generally encourages new growth, so shortening the branches will result in new stems emerging from each branch. This will probably give you 2-3 new branches on each of your existing branches. I suspect that the central leader will continue to grow upwards and will eventually develop new branches above your two existing ones. Time will tell.
Don't prune now. It's too late in the season and anything produced will be subject to winter dieback. Wait until early spring, just as the buds begin to swell on the branches. Cut them back by about one-third, trying to cut just in front of a bud scar. This is where new growth will develop. It may take several years of selective pruning, but eventually your mimosa tree should develop a full canopy.
Best wishes with your tree!
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