I would be tempted to do nothing for a couple weeks - basically to ensure the area will dry up.
in other words, the sprinkler leak you found and repaired was in fact the cause of the wet soggy conditions.
>>a partial scrap yard
sigh. common practice was to dig a big hole, dump in all the construction debris, put dirt over it.
well, the construction debris - mostly wood - rots away and yes that will result in the ground settling. sometime in the early 1980's the Federal EPA reclassified this practice as creating an illegal landfill. regrets, not every contractor/builder follows all the rules.
buried construction debris can take 10-15 years to fully collapse. when was the house built?
if I understand correctly, the leak that you found/fixed, it was on a leg of an individual sprinkler head. this means it would only be pressurized / leak when the sprinkler was operating. not knowing what your watering schedule is, I question whether that leak could saturate a 12 x 8 area to soggy/squishy in the 20 minutes +/- the sprinkler was pressurized. it's certainly possible, but it raises the question of whether all the leaks have been found.
which leads me to the "see if it dries up" waiting period - as obviously anything you do may have to be re-done should the area not dry up....
of course, if you're planning a wedding on the front yard next week, the option to "do nothing" may not fit the schedule.....
some sprinkler systems are installed with a house water supply pipe to a "box" in the yard - the box has the electrically operated valve controlled by the timer. this is most common for systems with multiple zones - a 25x20 front yard is likely only one zone. the water pipe from the house to the box is pressured all the time and if damaged will leak 24x7. there are other designs where the solenoid valves are above ground.
the fact that your ten inch hole is not collecting water is a good thing (presuming the system is energized / in operation.)
as for filling up the area - around here we get three "types" of dirt -
fill dirt - stones / rocks / whatever
screened top soil - basically dirt that has large chunks of non-dirt sifted out
(alternative: limed & screened - probably not common in your area)
"garden soil" - screened & limed top soil mixed 50-50 with compost.
for a grass area I'd go with the topsoil option. the "garden soil" will settle much more than "pure dirt" as the organic matter rots away.
if you have an area to "store" extra dirt, order more than you need as you'll like have to 'refill' the area after it settles.