Short answers, pulled from the story.
The Third Battle of Kharkov took place from the 19th of February to the 15th of March 1943. The German forces under Field Marshal Erich von Manstein launched their counterattack on the 19th of February, and the SS Division Leibstandarte completed the recapture of Kharkov on the 15th of March.
Field Marshal Erich von Manstein commanded Army Group South and directed the overall counteroffensive. General Paul Hausser commanded the SS Panzer Corps, and General Hermann Hoth led the 4th Panzer Army. Hausser famously defied Hoth's orders and launched a direct assault on Kharkov rather than encircling it from the north.
The Red Army suffered more than 80,000 personnel casualties during the Donets Campaign. Of those, an estimated 45,200 were killed or went missing, while another 41,200 were wounded.
Soviet forces had severely over-extended themselves during their winter offensive, leaving divisions far below authorized strength. Some divisions in the 69th Army could count on only 1,000 to 1,500 soldiers, and certain units had as few as 20 to 50 mortars. At the point of contact, Manstein's 350 tanks outnumbered Soviet armor almost seven to one, and German forces were far better supplied with fuel.
The SS Panzer Corps suffered approximately 4,300 enlisted personnel killed and wounded, along with around 160 officers, by the 17th of March 1943. This represented an estimated 44% of the corps' fighting strength, reflecting how costly the four days of house-to-house combat proved even for the attacking German side.
The German recapture of Kharkov on the 15th of March and Belgorod on the 18th of March 1943 created a salient in the Soviet lines. Hitler chose to attack that salient directly with Army Groups South and Center rather than wait for a Soviet offensive, a plan known as the "forehand method." That decision led directly to the Battle of Kursk in July 1943.