what is the distance between badaling and forbidden city?

Protima, it's about 70 kilometers. I personally think you should visit the forbidden city first because it is closed early (at 16:10). You know, it is opened at 08:30. You can take subway line 4 to get off at Lingjinghutong, get out at Exit B and keep walking north for 500 meters to Gangwashi station. There you can take no.109 bus to arrive.

You can spend around 2.5 hours inside and then exit from the north gate (Shengwu Gate) to take no.685 bus to reach Xizhimen. There you can walk a few minutes to find Beijing North Railway Station. You can take S2 train departing at 12:42 to reach the wall. It will arrive before 14:00. You have at least 2 hours to climb. In return, you can take S2 train too.