Who were the 13 persons who voted to choose PM among S Patel and JL Nehru? how many votes did Nehru get?
I don't think that a vote took place.
In the 1946 election for the Congress presidency, Patel stepped down in favour of Nehru at the request of Gandhi. The election's importance stemmed from the fact that the elected President would lead free India's first Government. Gandhi asked all 16 states representatives and Congress to elect the right person and Sardar Patel's name was proposed by 13 states representatives out of 16, but Patel respected Gandhi's request to not be the first prime minister. As the first Home Minister, Patel played a key role in integration of many princely states into the Indian federation.
And India TV has a longish article on the issue, which clarifies
. . . . the nominations were to be made by 15 state/regional Congress committees. Despite Gandhi’s well-known preference for Nehru as Congress president, not a single Congress committee nominated Nehru's name. On the contrary, 12 out of 15 Congress committees nominated Sardar Vallabh Bhai Patel. The remaining three Congress committees did not nominate any body’s name. Obviously, the overwhelming majority was in favour of Sardar Patel.
A disappointed Gandhi gave into Nehru's obduracy and asked Sardar Patel to withdraw his name. Sardar Patel had immense respect for Gandhi and he withdrew his candidature without wasting any time. And it paved the way for the coronation of Pandit Jawahar Lal Nehru as India’s first Prime Minister.
I don't understand why Wikipedia says that the nominating committee had 16 members, while India TV states that the committee had 15 members.
I don't know which congress committees proposed Patel, but I can speculate based on the following two quotes from wikipedia
In the elections, the Congress won a large majority of the elected seats, dominating the Hindu electorate. But the Muslim League led by Muhammad Ali Jinnah won a large majority of Muslim electorate seats. The League had resolved in 1940 to demand Pakistan—an independent state for Muslims—and was a fierce critic of the Congress. The Congress formed governments in all provinces save Sindh, Punjab and Bengal, where it entered into coalitions with other parties.
During his lifetime, Vallabhbhai Patel received criticism of an alleged bias against Muslims during the time of partition.
If Patel had a reputation for an anti-Muslim bias, then it seems probable that the states that did not submit his name were the majority Muslim states.