Plane the face of the bench top. File flats on two washers to make them 'D' shaped. The D washers must rest flat against the cast steel vice jaw. Install the complete vice and replace the leg assembly. Cut two pieces of scrap 1-by hardwood, oak or maple, a little larger than the vice jaws.
Hardwood flooring scrap works well.
These faces should stick up high enough to be planed with the top. The fixed jaw usually has threaded holes for bolts to attach its jaw face, and the moveable jaw has screw holes from the outside. Fit the top self to fill the room from the bench top to the back legs and screw it to the cross braces. Remove the back piece, round its corners, and screw and glue it in place. Drill a series of 1 inch holes straight back from the vice dog for a bench dog.