Is the following sentence grammatically correct? “What I wish to know is that whether the government is aware of the problem.”

No, it is not. In this case, you do not need “that”. Avoid using “that” and “whether” together.

*What I want to know is whether you can come to the party. *Ayesha wanted to know whether she had been admitted to the programme.

Source: ‘Know Your English’ Series – The Hindu Daily, July 25 , 2005