Lots of Options
I want to fish Eagle Creek this year, but getting there is either a 5+ mile hike each way to the mouth of the stream or a $50 ferry ride across the lake ($5/mile....). If you have the time and the energy, it could be a great trip.
As Craig mentioned Forney is another great option. Its about 5miles to campsite 71, and 72 is only 1.2 miles further, with very little change in gradient. Coming in from the top (Clingman's Dome Rd and Forney Ridge Trail) to campsite 68 is shorter (3.5 miles) but is much steeper (2000' elevation change), but could be great brookie water.
Noland Creek is another possibility, but I'm not sure how good the fishing really is. I have been there a couple of times and have not done as well as other places in the park, but the trail is great and the stream is beautiful.
As Charlie mentioned, Deep Creek is a great place as well. The stream is pretty overgrown up high, but the fishing can be outstanding. I would stay below campsite #53, it was in pretty bad shape last year. There are numerous campsites on Deep Creek and while it gets more pressure in spots than Eagle and Forney, its still pretty remote in the upper and middle sections.