no simplify option for expression written under Defer[] enviroment

Question

I am new to Mathematica. I have defined following functions

p[k_, j_, n_] := 
 9 2^11 2^(6 j) k^6 - 3 2^6 2^(4 j) k^4 + 5 2^2 2^(2 j) k^2 - (n + 1)
q[j_, n_] := 
 Defer[1/(24 2^(j - 1) Sqrt[2]) Sqrt[
   1 - 29/(604 + 648 n + 9 Sqrt[4805 + 9664 n + 5184 n^2])^(
    1/3) + (604 + 648 n + 9 Sqrt[4805 + 9664 n + 5184 n^2])^(1/3)] ]

In fact, q is a root of the polynomial p and I have checked it on other software. But the problem is that when I am evaluating p at q, I don't get zero. Furthermore, the simplify button disappeared from my screen in .nb file.

Here is my code:

p[q[j, n], j, n]

And I am getting the following answer:

-1 - n + 5 2^(
  2 + 2 j) (Sqrt[
   1 - 29/(604 + 648 n + 9 Sqrt[4805 + 9664 n + 5184 n^2])^(
    1/3) + (604 + 648 n + 9 Sqrt[4805 + 9664 n + 5184 n^2])^(1/3)]/(
   24 2^(j - 1) Sqrt[2]))^2 - 
 3 2^(6 + 4 j) (Sqrt[
   1 - 29/(604 + 648 n + 9 Sqrt[4805 + 9664 n + 5184 n^2])^(
    1/3) + (604 + 648 n + 9 Sqrt[4805 + 9664 n + 5184 n^2])^(1/3)]/(
   24 2^(j - 1) Sqrt[2]))^4 + 
 9 2^(11 + 6 j) (Sqrt[
   1 - 29/(604 + 648 n + 9 Sqrt[4805 + 9664 n + 5184 n^2])^(
    1/3) + (604 + 648 n + 9 Sqrt[4805 + 9664 n + 5184 n^2])^(1/3)]/(
   24 2^(j - 1) Sqrt[2]))^6

I know the above quantity should be equal to zero, but Mathematica is not simplifying it to make it zero. I used Defer[] in my expression, as I don't like the way Mathematica reorders my expression. Can somebody tell me how to resolve this issue?

I have tried

Simplify[p[q[j, n], j, n]]

and

Simplify[%]

But nothing is working.