A diagram can be helpful.
The direction of the ship's travel and the reference line for bearing measurements are parallel, so the interior angle of the triangle is the same as the easterly angle of the first sighting, 12°. The second sighting angle is 13° farther east (=25°-12°), so that is the measure of another of the angles of the triangle.
With the given information, we can use the Law of Sines to find the measures of the unknown sides of the triangle.
The long side is opposite the obtuse angle.
(long side)/sin(155°) = (7.8 mi)/sin(13°)
long side = sin(155°)·(7.8 mi)/sin(13°) ≈ 14.7 mi
And the unknown short side is opposite the 12° angle.
short side = sin(12°)·(7.8 mi)/sin(13°) ≈ 7.2 mi
The ship was first sighted 14.7 miles from the lighthouse.
The new location of the ship was 7.2 miles from the lighthouse.