how to glue the broken plastic pieces that dovetails the faucet valve
You should look with great care and bright light for some indication of a brand name somewhere on the faucet. Because a replacement part is really the only viable solution (short of replacing the whole faucet.)

1: Plastic glues are almost always not as strong as the original part. IF the plastic is a solvent-weldable plastic and IF you have the correct solvent, you have the possible exception, if you do a good job solvent-welding it. 

2: The original part broke, probably in part due to the plastic aging poorly - so if you do successfully glue it, it will probably break again, either in the same spot or a new spot.

Another option, depending how difficult it is to model, would be to have a new part 3D printed. But if you can buy a replacement (which starts with sorting out who made the faucet) that will be considerably less effort (unless you have someone that's dying to work cheap to show off what their 3D printer can do.)