"insistent need" is correct and usable in written English.
You can use it to refer to a urgent and strong need for something. For example: "He felt an insistent need to protect his family from harm."
The contribution to this year's carnival from Tobago's most successful contemporary soca artist, Shurwayne Winchester, is a song about a woman's insistent need for "the sweet Tobago love" – neatly combining two of soca's most popular themes, sex and patriotism.
As a result, there is an insistent need to investigate various dimensions of women entrepreneurship.
Her pursuit of the possibly dangerous Sammy constitutes the thin plot of the film, which is really a snapshot of Lila's ear-ringingly insistent need to touch and be touched.
Jenny, with her far more profound potential to reveal, offers no why or how either; aside from her insistent need to be left in prison for life, she, too, is mostly unreadable.
I guess I was expecting something fried or chargrilled, as they've done with the smoky sardines, the meaty, insistent flesh needing nothing more than a spritz of lemon.
Of all the cabinet ministers he was the most insistent on the need to resist Germany.
But Darling was very insistent about the need to go further to increase the supply of credit to business.
Compared with many rivals, the Swiss theologian's reading of Islam is more devout, and also more insistent on the need for Muslims to integrate in the West.
No one was more insistent on the need to keep out of a European war than the paper's chief leader writer and deputy editor, CE Montague.
Through it all, Ms. Maloney, who represents parts of Manhattan and Queens, remained defiant, insistent about the need to give Democratic voters a choice and pledging to run an aggressive campaign.
The real Twitter revolution may have been to make us better at talking to and reaching each other, and more insistent about the need to do so — and not to make us dependent on Twitter.
