The mulberry is renowned for being weedy and hard to kill. At my nursery we have them coming up in a tree line and they are impossible to kill (unfortunately the neighbors complain when it gets messy looking).
You
should able to chop it down to nothing and it will resprout.
The trick to training non-dwarf trees to being dwarf is to divert the
energy that would normally go into growth. Similar to pruning fruit trees, you want to get the branches to grow sideways so they are almost parallel to the ground (not completely though). The more horizontal branches are, the fewer hormones are released by the growing tip of the branch that say 'GROW!'. What I do to get branches to grow laterally is make 'limb spreaders' with a dowel (usually 12-16" long) and nails to push the limb in question down away from the trunk and/or clothespins with rocks tied to them.
Create a basic structure of lateral branches that you can prune back to every year or 2 (with some removal of very vigorous shoots as needed), and you should be able to maintain it to a smaller size. If you look up pruning fruit trees, check out pruning peaches and central leader pruning of apples, both should give some ideas on what you are trying to do.