Kumar exploited the seaming conditions to return figures of 4/46 and leave England struggling at 219 for six at close of play on an absorbing second day, still trailing by 76 runs.
Ballance struck his second century in four matches to help rescue England on a Lord's wicket that appeared to have eased out a little.
He scored 110 runs (203 balls, 15 fours) to help his lift his side from 113/4. For India, Kumar was the stand-out bowler and delivered 23 overs.
India had heaped up pressure on England and reduced them to 125/4 at tea, from where Ballance (51 batting) and Moeen Ali (32 runs, 106 balls, 4 fours) began in the last session. They put on 98 runs for the fifth wicket and tested the patience of the Indian bowlers.
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To their credit, the visitors didn't lose their focus and continued to bowl probing line even as the green-top pitch eased out. The medium pacers held tight lines and didn't allow easy runs. With the older ball getting soft, skipper MS Dhoni cleverly used Jadeja as well as Vijay.
His first century at this ground came against Sri Lanka just three matches prior. He fully utilised the life given to him by Dhoni who let go an edge induced by Stuart Binny (0-45) in the 40th over. Ballance had been on 32 at that time.
With two overs to go for the new ball, Ali finally got out to a lazy shot off Vijay, trapped LBW giving the opening batsman his maiden Test wicket. Plunkett was sent in as night-watchman but the ploy failed when Ballance was dismissed by Kumar in the 83rd over, caught behind by Dhoni who didn't make a mistake this time.