The problem with fusible is that it softens every time you iron over it, so eventually it un-repairs itself.
I would probably put the fusible in (but not fuse it yet), then hand-stitch the edges together, then fuse to reinforce the stitching.
I might also consider just re-covering the whole thing - cut fabric bigger than the ironing surface, hem the edge along the centre (where the mat/ironing surface folds), and wrap around and duct tape to the back on the other three sides. The best method is probably the one that suits you best aesthetically.