My daughter;s passport is slightly damaged

dear all,

my wife and daughter are in india. today morning by mistake my daughter''s passport got slightly damaged as it was left in her jeans and then into washing machine.
in the passport, the signs and stamps are readable and fine, no problem.
even the first and the last page are totally readable and fine.
but the very front page , which sticks to the black cover of the passport has been stuck out ( not separated, not torn )

We contacted kuwait embassy at india, to which they told that they have no problems at all, but they told to contact indigo airlines ( travelling by indigo on this 10th to kuwait ) and also at immigration
my family is trying to contact indigo and they are presently in Delhi.

But when she asked how to contact immigration, then they told that she has to go the airport and contact the enquiry.
the problem is that at delhi airport where to confirm with enquiry and what if the enquiry says ok and the immigration creates a problem.

Please help me to solve this matter as my family will travel from Delhi to kuwait on this 10th.

Should I contact indian embassy here or try something else ?