I day hiked into Jenny Lake once, and went off-trail to Weaver Lake via Shell Mountain. Until you posted, I didn't realize it is not in the National Park. I'd call and ask about permits:
Hume Lake Ranger District, 559-338-2251
35860 East Kings Canyon Rd, Dunlap, CA 93621
Monday through Friday, 8:00am - 4:30pm
When I hiked there, I thought Jenny Lk would make a fine camping site. Nice lake, nobody there. Weaver lake was more populated, with people swimming and playing in the water. I doubt fish are planted, so fishing could be sparse.
Here's a map:
Jennie Lake in the center Jennie Lk. would make a great base camp to make some nice day hikes farther into the back country.
