I do it using the same technique to fix wire insulation. Glue the parts together and then get the same type of plastic and melt it around the cracks. You can use an old iron to do that. Melt it good, no matter how bad it will look. then have a 400 sandpaper to remove the excess and a 1000 sandpaper to finish. After that use silicone to renew the plastic. It's gonna be like new.
In Brazil we have a low viscosity "super glue", like Super Bonder or "Crazy Glue" but stronger, it's name is 725 (basically some solution of ethyl cyanoacrylate), use something like that to hold the parts together before melting the plastic. Clean all the parts very well before applying the glue.