If the kids (and the car!) can handle the bumpy ride to the trailhead, this short out-and-back is totally doable!
Park Road 19 (also called Herdina Park Jeep Road) is very rough and requires a 4WD vehicle. Note: bikes are not allowed on this trail.
To reach the remote Eye of the Whale Trail, turn north onto Willow Flats Road (just across from the Balanced Rock parking area). Take Willow Flats Road (also called BLM378) for less than a mile, then turn north onto Park Road 19 (also called Herdina Park Jeep Road). Stay on Park Road 19 for 2.2 miles until you approach the first large rock formation and see a parking area to the west. This is the trailhead for Eye of the Whale Trail.
From the trailhead, the arch may not seem that noteworthy and won't really resemble a whale yet. Pass through the arch on this mellow 1/4 mile path, then turn around to look backwards. From the backside it is easy to see the heavy brow and eye hole of the whale.
If you liked your adventure to this area, seek out an unnamed arch accessible from this same trailhead. From the parking area, head around the north side of this rock formation for 1/4 mile and look for the arch nestled in the rock fins.