Hi, I'm sorry your learned Lily has such an injury as a puppy.  I got a similar diagnosis for my Nyaya (she has since passed) when she was 10 weeks old that you did for Lily .  Nyaya's injury was a severed nerve (she could not move or feel her leg from just below the shoulder down).  The first hope was for the nerve to grow back, if that didn't work then fusing the joint, then amputation.  The nerve failed to grow back and we learned the injury damaged growth platlets in the shoulder so amputation was our only option.  We did that when she was about 4 months old   Nyaya lived 14 and half years as a 3 legged dog.  She had a active happy life (fetching ball, playing Freesbie, swimming and frolicking in the snow), but it would been better if she had been able to keep her leg.  She was just shy of 15 when we let her go (failing liver and debilitating arthritis). 

I suggest you do what you can to help Lily keep her leg as it is for as long as possible, then progressively do the least invasive option needed to delay doing something that will negatively impact her mobility for the rest of her life.  Resort to amputation only if/when there is no other choice.  But if you get to that point rest assured your dog's life will not be ruined by having to have it done and that a 3-legged dog can live a long happy and active life.

Take a long view on Lily's life.  Think about how what she does today will impact how she feels when she is over 10.  This will mean balancing the immediate benefit to her in the now against out the impact of that activity on her in the long term.  Playing is a must, but you will need to step in and call a time out when she starts looking exhausted.  Instead of playing hard for an hour break it up into three or four 10 minute periods.  Consider crate training her as when she is in the crate you know she is not overdoing it.  Nyaya loved her crate and it was a way we could control her activity when she was a puppy (she was fed in it and always got a treat after entering, it became her safe place). 

Take her swimming regularly if that is at all possible. Strong muscle compensates for weak joints and swimming is a wonderful exercise. You can let her swim for as long as she wants/can keep her head above water.  Commit to keeping her in shape over her life.  Extra weight is not good for damaged joints.  

If it is a front leg try to prevent her from pounding the remaining front leg.   I tried to prevent Nyaya from doing things like running down stairs or leaping out of the car.  I carried her down the stairs and then put a baby gate up to prevent her from running up and down them through the day while I was done.  I did let her run down hills and leap from the couch when the door bell rang as those things are necessary for a happy dog life.  But I put an old cushion from a recliner on the floor below the couch so she would be leaping onto it before hitting the floor.  

Given I was moderating Nyaya's activity I didn't 'use up' her active time on dull walking through the neighborhood.  I drove there or pushed her in a cart to the park so 100% of her active time was used for 'fun stuff'.  If you like biking you can get an attachment that will let you pull Lily with your bike.  In the park we visited there was a man who had trained his dog to jump into the cart before they got onto a the road and jump out when there were in the park (these carts have zippers if you dog is not so obedient).

You said watching her activity level was 'impossible'.  It is not impossible you just have to be creative and diligent.  I have imagined that having a dog that I didn't have to constantly monitor and manage to prevent them from injuring themselves must be pretty cool.  But needing to do so is not that big a deal and I am thankful that I did everything I could to make Nyaya's life fun and active for as long as it was.