Thanks, I watched a YouTube video of the stock vinyl going wrong and I knew that was beyond me.
Since you had broken corners, I would use auto body filler, and sand them till they are flat and square corners before trying to veneer them. Paperbacked veener would have been easier, but it is more expensive. Apply the glue to one side of speaker, than dry glue with hair dryer or gently with a heat gun till glue is transparent, or just wait if you are patient.
The raw wood veneer would want to curl up and become unmanageable with glue applied. What I found worked well was using a sewing cutting mat, covering that with parchment paper, then putting the veneer on top, then tape the veneer and parchment paper to the cutting mat. Apply glue evenly with a putty knife or glue roller, then gently dry glue with heat gun or hair drier till it just becomes transparent.
Then turn the speaker over and place on top of the veneer. Press down and then trim excess off with carpet knife or utility knife as close as you can without cutting into it. Like 1/8} extra is good target. Then flip over after all 4 sides trimmed, and iron on with gentle heat with parchment paper between iron and wood. Paper keeps glue from gluing to table if there are wholes, and protects wood during ironing. Cotton setting caused splitting, I think I did wool or silk, then let dry for 30 minutes then trim remains edge with veneer trimmer, or just sand flush with a sanding block carefully to not sand through corner, or router with trim bit.
Once finished and complete, find the pre drilled holes and drill the veneer, and drill wide just through veneer layer, or screws will try to lift veneer.
Glue. I used grill a gorilla glue carpenter's glue. In theory this can loosen with age, but will probably be fine for these small speakers. There is glue that is made specifically for the iron on method. I think it is called hot press. Do not, I repeat do not use cold press glue with iron on method, it takes too long to dry, and does not tack well. Hide glue works well, but needs to be heated up in water bath first, will slide around while wet, and it really needs a day or two to cure before doing the next panel. I reinforced my screw holes with CA glue (superglue).
Picture below is what veneer looked like with glue applied, and dry to touch, ready to mate with other dried to touch glue on the cabinet. Trimming the veneer for the openings was done with a trim bit in a router, but the recessed areas had to be trimmed slowly and carefully with hobby knife and then used miniature cylinder drum sanding bits in drill to sand to edge. This took 3 weekends to veneer and finish, probably should have taken 4 weekends to not rush. I made lots of mistakes along the way.