The new software 13.6 is downloaded but will not let me install

13.6 is no longer available to install. Its been replaced by 13.6.1 and as such can no longer be installed.

Go to Settings->General->iPhone Storage, find the 13.6 installer and delete it. Then go to Settings->General->Software Update, and let 13.6.1 download and install that.

Are you getting an error message? Is the Install button just greyed out?

and does it tell you what's required to update? Typically it will tell you something like.. your phone needs to be plugged in and connected to WiFi