There are a few choices available for you. Los Angeles was a Baltimore class heavy cruiser. Within that class, there were two sub-groups, one with a square-ish stern, with two aircraft cranes, and one with a rounded stern with a centerline crane.
Without getting into more exotic and much more expensive resin kits, there are a couple of plastic kits that should be reasonably available. (eBay, if the regular online dealers don't have them anymore.) Revell made one at about 1/480 scale, about 14"-15" long as I recall. Over the many years they produced the kit, the box top may say Los Angeles, Pittsburgh or Helena - all the same plastic inside. It is not a museum quality kit, but may be a good place to start.
Trumpeter has made the kit in 1/700 scale, just under 12" long. The smaller Trumpeter kit has many smaller pieces, and if you don't mind working with smaller parts, it can look really good. They make it with both the square stern, labeled as USS Baltimore, and the rounded stern as USS Pittsburgh.
During the time your Dad was aboard, the only significant difference from the WWII appearance is the removal of the aircraft catapults at the stern because she transitioned from carrying seaplanes to helicopters. The single aircraft crane remained.
There is a booklet called "USS Los Angeles (CA-135), Cold War Sentinel", by Harvey M Beigel, with about 60 pages and a number of nice photos. It shows some photos of the gunfire damage to the ship from North Korean batteries in 1953.
You can also find photos and a link to the official ship's history at www.navsource.org/.../04135.htm.
Good luck,
Rick