Depends on your priorities. Looks? Function? Strength? All three?
Personally I would replace the handle. To do that you'd need to drill out the rivet on each side (assuming it is a rivet, if its a chicago screw it is considerably easier, just unscrew it), Clip the stitches. Once that is done you can either make a new handle or buy a premade one like those pictured above. If you're making it you'll need leather (moderately thick like 4-6oz, probably on the low end of that) some barge cement, some cord, a couple of rivets or chicago screws, a couple of D rings, either a pair of needles or a speed stitching awl, and some waxed nylon thread, and if you so desire leather dye.
Cut the leather into about 1" strips the length of the handle you want plus about 4 inches, Cut each end to a point about a half an inch long.. Apply barge cement to the flesh (rough) rough side of the leather, allow it to dry. Apply a second coat. When the second coat is tacky lay the cord in the middle. Fold the leather over the cord pressing the edges together leaving about 4" at each side not folded. Put the round end of the D-ring in the middle of the unfolded portion and fold the leather back over the D-ring so that the point nestles into the crook of the other fold. Using a stitching chisel or a scratch awl punch holes along the edges of the handle. Stitch along the entire edge finishing the handle. Now you need to make tabs to attach the handle to the bags. For the tabs cut more leather into 3 to 4 inch strips pointed at both sides. Cover the tab in barge, let it dry until tacky, fold it around the flat side of the D-ring, punch stitching holes around the outside of the tab, and a larger hole in the middle for the rivet or chicago screw. And then rivet/screw and stitch it to the bag.
If you're not up for replacing the handle you'd want a wrap around patch that is attached to both the remainder of the base part and the handle part with both stitching and barge contact cement. It won't look as clean, or be as strong as a replaced handle but it'll get the job done.