They make special glue for plastics which bonds the pieces together and looks invisible. It has a special applicator that looks like a needle so you can get it in between the pieces you are joining. I would call your local glass or plastics company and ask them which type of glue you need. If you try to use a wrong type of glue you could get it on a part you don't want and it can distort or put a film of glue on it which you won't be able to see through.

And take it SLOW when you are drilling or cutting the plastic, if you go too fast you can melt it or even worse crack or chip the plastic. Lexan is usually the best type of plastic to use it is more durable and won't chip or crack. Lexan is what they use in car windshields to help prevent shattering.