I am having a trouble searching for public transportation from Mutianyu to Tiananmen.

Firstly, I'd like to tell you bus no. 936 is still in service and it operate between Dongzhimen and Jiugukou. Jiugukou is still around 30km to the great wall and there is no direct public transport leading there.

Personally, I suggest you take subway line 1 at TAM Square East Station to Jianguomen and transfer to line 2 to Dongzhimen. You can take no. 916 or 916 express line nearby to Huairou Bus Station or Huairou North Street. After that, hire a minivan to Mutianyu in 20min and the cost is CNY40-50.