On mine the reverse gear takes several seconds to pressure up and work. How long have you waited for it to work when put in reverse? If the 0-ring has a crack or chip in it the it will let pressure pass through before it builds up and holds for the gear. If after say 10 seconds you still do not have reverse gear then the o-ring is gone and you won't have reverse. And if it is the with 115k on it you might as well have the tranny rebuilt or replaced with one from a salvage yard. If rebuild or replacement is not an option then you could just make sure you don''t park where you will have to back up. It could drive for days or years this way.
The miss could be a bad spark plug to a vacuum leak.