Here's the pertinent stuff on the South Dakota from an article I wrote:

Savo Island, 13 Nov 1942. This battle along with North Cape below represents the only two actions that occur at night. As such, they highlight the sensory revolution that occurred during World War 2. This particular action had several parts to it. For purposes of this paper, only the battleship action need be discussed. Starting at 2300 the two US battleships, Washington and South Dakota obtained radar contact on the various Japanese ships operating just south of Savo Island. First fires were directed from both battleships against the Japanese cruiser Sendai and destroyer Shikinami without success between 2316 and 2319. Range was 18,500 yards decreasing to 12 to 13,000 yards. Washington fired 42 16" rounds and 100 rounds of 5". Interference from Savo Island and other clutter made getting good radar resolution on these targets difficult resulting in the poor results.

At 2333 the South Dakota suffered an electrical failure in #3 secondary fire control director. Either the crew of this director (likely) or one of the ship's electricians locked in the tripped circuit breaker to restore power. This action caused a cascade failure of the main bus tie breaker from which this system was powered. As a result, the South Dakota lost electrical power to most of her superstructure including all radar systems. The ship's gyro and central fire control system were also lost in this failure.

The ship's electricians traced the fault and corrected the problem almost immediately but, the fault reoccurred when this same director switched to their alternate power supply reinserting the fault into the electrical system. Electrical outages and problems continued to plague the South Dakota as part of this fault. While the fault was finally corrected by 2336, the result of this was that South Dakota was partially blinded and distracted by these electrical problems.

Both battleships continued to close with the Japanese forces. At 2335 the South Dakota made a turn to port to avoid the wreckage of the destroyer Preston and in doing so began to close more rapidly than the Washington with the Japanese. With power restored, the South Dakota reopened fire on the Sendai at 2342. The initial salvos set the aircraft on her fantail on fire (these were subsequently blown overboard by additional salvos). South Dakota was now silhouetted from behind by the wreck of the Preston and had a fire burning aboard that allowed the Japanese main body to see her. Washington meantime was still maintaining her original track, closing with the same Japanese ships.

At 2355 the Japanese launched a 34 torpedo attack on South Dakota at a range of about 6,000 yards. No hits were scored but, had some been made this might have significantly changed the outcome of this battle.

Three minutes later the Kirishima, Takao and Atago all opened fire on South Dakota as well. Either the Kirishima or one of the cruisers illuminated the South Dakota with a searchlight. The range was now down to 5,000 yards.

At 0001 the Washington opened fire based on her radar plot with optical verification on Kirishima. In 6 minutes Washington fired 75 16" and 107 5" rounds hitting the Kirishima with 9 16" and about 40 5" rounds. Her radar plot was able to track individual 16" shell splashes and hits. These hits wrecked the Kirishima, setting her afire and destroying her steering gear. Few, if any, of these hits were on the hull or below the waterline. Kirishima 's engines were intact even as the ship had major fires burning in the superstructure.

Washington also took the two Japanese cruisers under fire with her 5" guns firing 133 rounds at one or the other at ranges from 10,000 to 12,000 yards.

South Dakota suffered 42 hits on her superstructure. Of these only one was a 14" shell. Kirishima took at least 9 16" and 40 5" hits as previously noted also all on her superstructure. The difference between the two outcomes is stark. In both cases the large number of small shells had little effect on either ship's ability to fight. It was the nine 16" hits on Kirishima that wrecked her. These apparently knocked out her main battery, her secondary battery, wrecked the steering gear, and started serious fires that eventually caused her abandonment. Of note is that both ships suffered virtually no hull damage due to the short range.

Some additional information: These shells did knock out all but one radar on the ship. The main and secondary batteries were uneffected and operational. Most of the damage was very superficial. The command staff was uneffected in the conning tower and the ship was never out of control.