yes, the job is too small, I am sure you heard that. But, like me, I take pride in the small jobs, because small job people have bigger jobs later on. Here is a person with a home problem and it needs addressing.
First, I would relieve the stress on the gutter by removing all the debris and weight. Depending on the gutter height, I would use an 8 foot A-frame ladder.
I would suspect that the nails or clips have worked their way out of the facia board and just barely hanging on. If this has been like this for awhile, I would look at the condition of the facia board. If exposed to dirt and water, it could be rotten and not able to hold a good nail or screw. You must have sound wood in the area to even begin fixing the problem.
You have another situation that the "fall" or level is off. Meaning, each run of guttering is at a small angle toward the nearest down spout. Water runs with the assist of gravity. This also has to be addressed or the water will again pond in the area and cause the same thing to happen again.
Create the proper "FALL" and use those handy screw on clips that attach the gutter to the facia board. I would attach one every two feet and replace the other fasteners in and around the defective area just to make sure.
As Jay stated, if there is a joint there, be sure and use a good caulk to seal the joint. I prefer the material called Butyl Rubber Sealant. it is good and does last.
Good Luck on your project and I feel that with assistance and guidance, you two can fix this problem.
Get back with us if you have any new SNAFU's while on the job.