How to secure privacy lattice to top of existing fence gate?
My first choice would be rebuilding the gate with continuous stiles from bottom to top and diagonal brace running from the rail below the lattice to the bottom. You can reuse the boards and hardware, so it won't require extensive amounts of new lumber and will match the existing fence from the outside.

It's hard to say without seeing the framed side, but I suspect any repair is going to be unsatisfactory in the long run. That lattice box is just too much mass to be unsupported. Sistering a second two by to the bottom of the lattice frame and finish nailing through the boards will prevent it from moving independently, but you're really asking for trouble if people are rough with the gate.