Although certain lakes are often known for certain hatches, there really isn't a "magic fly" for any lake, LLJ included.
Any flybox should have the following staples:
- chironomid (sizes 10-18, various colours and styles)
- damselfly nymphs (sizes 10-14, olives, greens, and tans)
- dragonfly nymphs (both big chunky ones and smaller skinny ones in various earthy tones)
- leeches (sizes 6-16)
- scuds (sizes 12-16 in various greens, blues, tans, etc)
- mayfly nymphs (sizes 12-16 in brown and tan)
- dry flies (elk hair caddis, mayflies, tom thumbs)
If you have the above flies, you'll cover 75-80% of the situations. On top of that, you'll need other sizes as well, and other flies eventually like boatmen, ants, etc. But the flies above will get you most of the way there. If it were me, I'd check the shoreline for what bugs are around, and if I don't find anything, I'd try chironomid or micro-leeches to start.
Good luck!