Play ended on day five with Kumar undefeated on 63 — after 58 in the first innings — when India declared its second innings at 391-9 in the final session.
India led by 352 by the end but had resisted the temptation to declare earlier and have an outside chance of causing England any concern.
Earlier Sunday, England had struck three times in the opening session as it looked to try and force a result which always looked unlikely on a lifeless and slow pitch.
"It is a tough job being a groundsman. But it was not ideal," England captain Alastair Cook admitted. "It was pretty flat and you were just trying to wait for a batsman error."
Stuart Broad provided England with an early breakthrough when he trapped Virat Kohli on 8. In his next over, Ajinkya Rahane (24) edged Broad behind to Matt Prior.
And Broad was unfortunate not to take the wicket of India captain Mahendra Singh Dhoni, who edged to his opposite number Cook at first slip on five, only for Cook to drop a straightforward chance.
But Dhoni did not take full advantage of his reprieve, adding only six more runs before Liam Plunkett bowled him with his first ball of the day.
Debutant Stuart Binny then came to the crease and guided India away from danger with his knock of 78, aided by 31 from Ravindra Jadeja whose innings ended when he edged James Anderson behind to Prior.
All-rounder Binny, who only made one run in the first innings and failed to take a wicket in his 10 overs bowled, brought up his maiden half century with consecutive fours off Plunkett. He had faced 86 balls.
"He has played a lot of domestic cricket and that exposure has really helped him," Dhoni said of Binny. "He is someone who can definitely contribute for a long time. This wicket did not suit him much but I feel as the series progresses, he will be a good bowler."
Alongside Kumar, Binny put on a 91-run partnership for the eighth wicket before Moeen Ali removed him lbw.
Kumar, who also got a five-wicket haul in England's innings, then brought up his 50 with a single off Ali to become the second player batting ninth to make two 50s in the same test after Australia's Peter Siddle, against India in Delhi in 2013.
Cook, who surprisingly bowled two overs to the delight of home fans at Trent Bridge, then removed Sharma for 13, who was caught behind by Prior before India declared for a draw.
It was Cook's first test wicket.
"It was a good test for us," Dhoni said. "We put pressure on them in their first innings. We tried quite a few things but it doesn't always go your way. It was difficult when the bowlers got tired and the ball got softer."
England's Anderson was awarded man of the match for his test record 10th wicket 198-run partnership with Joe Root, which saw him set the highest ever batting score by an England No. 11 of 81.
"I'm actually more stiff from batting, to be honest," said Anderson, who bowled 59 overs in the test match. "I've got a new found respect for batsmen. I think you've got to give credit to both teams on that pitch for trying to create entertaining cricket. We tried really hard today to force a win but it wasn't to be."
Cook also praised the efforts of Anderson and Root, who made an unbeaten 154, as they batted from 298-9 to 496 all out on day four.
"It was an outstanding hundred from Joe Root who has really come back to his best," Cook said. "For Jimmy to get 80, I didn't see it coming but thank God he got it."
The second test of the five-match series starts Thursday at Lord's.