Short answers, pulled from the story.
The Battle of Kiev began on the 22nd of June 1941 with Operation Barbarossa and concluded by the 2nd of October 1941 when most trapped forces were eliminated. The encirclement was completed between the 16th of September and the 17th of September 1941.
General Mikhail Kirponos commanded the Southwestern Front which held four field armies along the border during the battle. Stalin ordered his forces to hold Kiev at all costs despite clear strategic risks and refused to authorize withdrawal until the 17th of September 1941.
The Southwestern Front suffered 700,544 total casualties including 616,304 killed captured or missing during the battle. Only 15,000 men escaped from the pocket out of hundreds of thousands trapped inside by the 2nd of October 1941.
Panzer Group 1 led by General Ewald von Kleist advanced toward Romny while Panzer Group 2 under Heinz Guderian moved through Gomel. The full connection occurred on the 16th of September south of Lokhvytsia when the 3rd and 16th Panzer Divisions linked up.
Stalin refused to authorize a withdrawal from Kiev until the 17th of September despite repeated requests from commanders like Kirponos and Budyonny. Kirponos ordered his forces to attack eastward without written confirmation from Moscow and died trying to break out of the encirclement behind enemy lines.