I can't remember if I got this particular case when I played the PI career, but do remember a little trick for the stakeouts.
I don't think it matters where in the park you go to do the stakeout, however you do want to leave room for the action to take place. Or else the game says something like There's nothing going on here.
I had that happen a few times at the library. If needed, you can cancel the case. It will more than likely come back up later in the career.
There was a case I had to cancel, because I couldn't do the 2nd to last step of the case. The specific case came back up and I was able to complete it.