Looking for an inexpensive room in Hawaii ( preferably Kaui) from Dec. 16 to 30th for 72 year old couple in good health for
$135.00 USD per night total.

Over Christmas which is the very busiest/most popular time of year??? Your best bet will be airbnb/HomeAway/VRBO . . .

Many people make reservations for Christmas a year in advance. And in any case, $135 a night is pretty cheap for Hawaii any time of the year. I believe each of the islands has a hostel, you might check on that.

Just checked Aston Islander on the Beach which is one of the cheapest decent places on Kauai, for the week of Christmas it appears to be booked, but rates for that time are about $250 plus tax.
There is one hostel, possibly two, in Kapa'a.
Make your rental car reservations well in advance, all the islands run out at Christmas time.

You'll have a much better chance of finding this in Mexico than you will in Hawaii (per your other post).